A mobile manipulator state constraint structure keeps robust optimal control method

Abstract

The application discloses a kind of mobile manipulator state constraint structure keeps robust optimal control method, including by introducing barrier function, offset Hamiltonian is converted into transformation offset Hamiltonian;According to the linear extended state observer designed, estimated disturbance is obtained;According to the position error after transformation and transformation offset Hamiltonian, performance index function is constructed to obtain optimal value function;According to optimal value function, HJB equation is constructed;Approximate optimal control law is designed;Based on approximate optimal value function, HJB equation is combined to obtain simplified approximate hamiltonian;According to the simplified approximate hamiltonian function error of hamiltonian function is defined, to design the weight update law of neural network optimal feedback control law is obtained;According to the robust optimal controller designed, the structure of mobile manipulator keeps robust optimal control is realized.The application solves the problem that existing method cannot organically unify constraint processing and optimal performance guarantee under the premise of strictly maintaining system energy structure and passivity.

Description

Technical Field

[0001] This invention relates to the field of mobile robotic arm control technology, and in particular to a robust optimal control method for maintaining the state constraint structure of a mobile robotic arm. Background Technology

[0002] Mobile robotic arms, consisting of a mobile platform and a robotic arm, have attracted widespread attention from the academic community due to their superior mobility and operational capabilities. These advantages make them indispensable in missions such as disaster relief, industrial assembly, and planetary exploration. However, the strongly coupled dynamics, nonholonomic constraints, parameter uncertainties, and external disturbances inherent in such systems pose significant challenges to achieving high-precision trajectory tracking and robust control.

[0003] To address these control challenges, researchers have proposed various control strategies over the past few decades, including neural network observers based on classical models (which do not rely on exact models), sliding mode control, adaptive techniques, virtual decomposition methods, and general coordination methods. However, in-depth analysis reveals that many existing methods have limitations: whether model-based or data-driven, they often treat the system as a "black box" or simply rely on numerical approximation. This results in the designed controllers lacking an energy-based physical interpretation, potentially compromising closed-loop stability and predictability—especially when the system operates under stringent constraints or uncertain conditions.

[0004] In contrast, the port Hamiltonian framework captures energy flow through the Hamiltonian matrix, interconnection matrix, and damping matrix, providing a natural basis for stability analysis based on passivity. Within this framework, passivity-based interconnection and damping configuration control achieves stability through energy shaping and damping injection without the need to cancel out nonlinear terms. As a nonholonomic system, the mobile robotic arm is constrained by the non-integrable constraint between the platform and the arm's velocities, increasing the difficulty of control design. Although existing methods often rely on coordinate transformations that mask the physical structure, the port Hamiltonian framework preserves the system's energy distribution pattern. However, extending this framework to mobile robotic arms with time-varying constraints, uncertainties, and disturbances remains an open challenge. In practical applications, mobile robotic arms often face time-varying state constraints caused by physical limitations, safety requirements, or workspace boundaries. To address these constraints, various control strategies have emerged, including model predictive control, control barrier functions, barrier Lyapunov functions, and preset performance control. While providing optimal performance under constraints, methods are computationally burdensome and rely on accurate system models. Barrier-based Lyapunov function methods ensure constraint satisfaction by incorporating barrier functions into Lyapunov function design, but typically impose strict feasibility conditions on the virtual controller and require the constraint functions to be strictly positive or negative. To overcome these limitations, researchers have recently proposed a general barrier function method. A general barrier function can transform a constrained system into an equivalent unconstrained system without imposing feasibility conditions or sign restrictions on the constraint boundaries, thus providing a unified framework for efficiently handling various dynamic constraints.

[0005] Achieving optimal control performance while satisfying state constraints is a highly challenging yet crucial goal for mobile robotic arms. Traditional optimal control methods require solving the Hamilton-Jacobi-Bellman equations, which are not only difficult to solve analytically for nonlinear systems but also extremely computationally expensive. To address this, researchers have extensively explored data-driven methods such as adaptive dynamic programming and reinforcement learning, approximating the optimal controller online through policy iteration. While these methods are highly adaptable and model-free, they often suffer from slow convergence, high computational costs, and a lack of stability guarantees, especially in tracking tasks where the control input does not tend to zero. Furthermore, many existing robust reinforcement learning designs introduce conservative upper bounds on uncertainties in the cost function, leading to suboptimal performance. Although recent research attempts to circumvent the zero-equilibrium assumption and eliminate dependence on discount factors by designing novel cost functions that incorporate tracking errors and their derivatives, these methods typically focus on numerical approximation and fail to address the underlying physical nature of the system's energy flow. Meanwhile, although the port Hamiltonian framework has achieved significant results in system stabilization and passive control, its deep integration with optimal control design (especially in scenarios with time-varying state constraints) remains largely unexplored. Existing technologies cannot organically unify constraint handling and optimal performance assurance while strictly maintaining the system's energy structure and passivity; this constitutes a core technical challenge that still needs to be addressed in this field. Summary of the Invention

[0006] This invention provides a robust optimal control method for maintaining the state constraint structure of a mobile robotic arm to overcome the above-mentioned technical problems.

[0007] To achieve the above objectives, the technical solution of the present invention is as follows: A robust optimal control method for maintaining the state constraints of a mobile robotic arm, specifically including the following steps: S1: Establish a port Hamiltonian dynamics model for the mobile robotic arm; and the port Hamiltonian dynamics model includes unknown transformation perturbation terms; S2: Construct the desired motion equation based on the port Hamiltonian dynamics model; design a feedforward controller according to the desired motion equation; define the offset Hamiltonian amount according to the port Hamiltonian dynamics model and the desired motion equation; S3: Based on the port Hamiltonian dynamics model, a coordinate transformation strategy is constructed by introducing a barrier function to map the time-varying state constraints of the robotic arm to an unconstrained space; the offset Hamiltonian is converted into a transformed offset Hamiltonian through the coordinate transformation strategy. S4: Design a linear extended state observer, estimate the unknown transformation disturbance term based on the linear extended state observer, and obtain the estimated disturbance; S5: Construct a performance index function based on the transformed offset Hamiltonian; obtain the optimal value function based on the performance index function according to the Bellman optimality principle; construct the Hamilton-Jacobi-Bellman equation based on the optimal value function; S6: A neural network is used to approximate the optimal value function to design an approximate optimal control law; at the same time, a simplified approximate Hamiltonian function is obtained by combining the approximate optimal value function with the Milton-Jacobi-Bellman equation. The Hamiltonian function error is defined based on the simplified approximate Hamiltonian function; the weight update rule of the neural network is designed based on the Hamiltonian function error; the approximate optimal control law is updated based on the weight update rule, and the optimal feedback control law is obtained. S7: Construct a robust optimal controller based on the optimal feedback control law, estimated disturbance, and feedforward controller; implement the structure-maintaining robust optimal control process of the mobile robotic arm oriented towards state constraints based on the robust optimal controller.

[0008] Furthermore, the method for establishing the port Hamiltonian dynamics model of the mobile robotic arm in S1 specifically includes the following steps: S11: The dynamic model of the mobile robotic arm subject to nonholonomic constraints is established as follows:

[0009]

[0010] In the formula: Represents the generalized inertia matrix; Represents the matrix of Coriolis force and centrifugal force; Represents the gravity vector; This represents the active control input torque vector; Represents a nonholonomic constraint matrix; Represents the constraint force vector; Denotes the lumped disturbance vector and ; Represents the generalized velocity vector of the system first derivative and ; Indicates transpose; This indicates the velocity of the mobile platform along the X and Y axes in the geodetic coordinate system; This indicates the yaw rate of the preset mobile platform; This represents the joint angular velocity of the robotic arm link in a preset geodetic coordinate system; Represents independent velocity vectors first derivative and ; These represent the angular velocities of the left drive wheel and the right drive wheel of the mobile platform, respectively. This represents the joint angular velocity of the robotic arm link in a preset reduced-order coordinate system; Represents the uncertainty term in the model; This indicates external interference; Represents the intermediate parameter matrix; Let A be the null space matrix representing the nonholonomic constraint matrix A; S12: The reduced-order model obtained from the mobile robotic arm dynamics model is:

[0011]

[0012] In the formula: express The first derivative; Indicates intermediate parameters; express The first derivative; S13: Define the system state vector of the mobile robotic arm according to the reduced-order model. for:

[0013]

[0014] In the formula: Represents the position coordinates of the mobile robotic arm first derivative and ; Indicates the location coordinates of the mobile platform; This indicates the position coordinates of the end effector of the mobile robotic arm; Represents the kinematic Jacobian matrix of the system; Represents generalized momentum; S14: The Hamiltonian dynamics model of the mobile robotic arm, based on S13, is as follows:

[0015]

[0016]

[0017] In the formula: Representing the Hamiltonian function Regarding the system state vector gradient vector and ; This represents the potential energy of the system; This represents the antisymmetric commutative term in the Coriolis matrix after order reduction; This represents the damping coefficient components in the damping matrix; Represents the system state vector The input matrix and ; Represents the identity matrix; Represents the system state vector The parameter matrix; Represents the system state vector Unknown transformation perturbation term and .

[0018] Furthermore, S2 specifically includes the following steps: S21: Given the desired trajectory and Based on the Hamiltonian dynamics model of the port, the desired equation of motion is constructed as follows:

[0019] In the formula: express The first derivative; express The abbreviated form; Represents the desired position coordinates and desired generalized momentum of the moving robotic arm; Represents a given desired trajectory The input matrix; Represents a given desired trajectory The parameter matrix; Represents a given desired trajectory The unknown transformation disturbance term; Indicates a feedforward controller; S22: The feedforward controller is designed based on the desired motion equation as follows:

[0020] In the formula: express The first derivative; This represents the kinematic Jacobian matrix corresponding to the desired equation of motion; Let represent the partial derivatives of the Hamiltonian with respect to the position and momentum components, respectively; express The abbreviated form; express The abbreviated form; S23: Based on the port Hamiltonian dynamics model and the desired motion equation, the offset Hamiltonian is defined as:

[0021] In the formula: This represents the offset Hamiltonian.

[0022] Furthermore, S3 specifically includes the following steps: S31: Define the unconstrained state after the transformation as... and ; Based on the port Hamiltonian dynamics model, the time-varying constraints are defined as follows: ; These represent the upper and lower bounds of the time-varying constraint, respectively. Represents the original state The A constrained state component; And the transformed unconstrained state The formula for obtaining it is:

[0023]

[0024] In the formula: express The abbreviated form; Indicate design parameters and ; Represents the barrier function; S32: Based on S31, the transformed unconstrained state derivative Represented as:

[0025]

[0026]

[0027]

[0028] In the formula: express The abbreviated form; express The Middle One element; express The Middle One element; express The abbreviated form; Represents the parameter matrix; express The first derivative; express The abbreviated form; S33: Define the desired transformation state as and express The abbreviated form; based on S32 combined with the desired transformation state is Obtain the transformed tracking error and ; S34: Based on the definition in S33, obtain the derivative of the compact tracking error. for:

[0029] In the formula: This represents the tracking error vector of the system position components after obstacle function transformation; This represents the tracking error vector of the system's generalized momentum components after the barrier function transformation; This represents the drift term or nominal system dynamics term in the transformed error dynamics equation; Indicates the transformed disturbance and ; This represents the optimal feedback control input. S35: Convert the offset Hamiltonian into a transformed offset Hamiltonian using the coordinate transformation strategy, i.e., convert the offset Hamiltonian into a transformed offset Hamiltonian based on S34. for:

[0030] In the formula: Represents the Hessian matrix; The quadratic energy weight matrix represents the generalized momentum components in the transformed error space.

[0031] Furthermore, S4 specifically includes the following steps: S41: Design a linearly extended state observer as follows:

[0032] In the formula: express The One component; and Indicates the first The tracking error of a generalized momentum channel after barrier function transformation; Indicates extended state and Indicates the action in the Lumped disturbances on the channel; express The estimated value; express The first derivative; express The abbreviated form; Denotes the observer gain coefficients adaptively generated online by the RBF neural network; S42: Estimate the unknown transformation disturbance term using the linear extended state observer to obtain the estimated disturbance. and .

[0033] Furthermore, S5 specifically includes the following steps: S51: The performance index function is constructed based on the transformed position error and the transformed offset Hamiltonian as follows:

[0034] In the formula: Represents a performance metric function; Indicates the discount factor; The energy weight matrix represents the quadratic form of the position components in the transformed error space, and... , Indicates the weighting factor; This represents the diagonal matrix corresponding to the position in the desired trajectory; S52: Based on the Bellman optimality principle, the optimal value function is obtained from the performance index function as follows:

[0035] In the formula: Represents the optimal value function; Indicates a time constant; Indicates the discount factor; Represents the optimal value function Regarding the error state after transformation The gradient; Indicates the nominal dynamic item The generalized momentum component; express The optimal value function corresponding to the time step; This indicates that after the barrier function transformation, The tracking error vector of the system position component corresponding to the time step; S53: Construct the Hamilton-Jacobi-Bellman equation based on the optimal value function. for:

[0036] In the formula: Represents the optimal value function to be solved; Indicates the nominal dynamic item The generalized momentum component.

[0037] Furthermore, S6 specifically includes the following steps: S61: Solve the Hamilton-Jacobi-Bellman equations based on the Bellman optimality principle. And obtain information about Optimal strategy for:

[0038] In the formula: Indicates intermediate parameters and ; S62: Using neural networks to approximate the optimal value function to rewrite the optimal strategy for:

[0039] In the formula: This represents the function by which a neural network approximates the true optimal value. The residuals generated during the process; A vector representing linearly independent basis functions; Represents the weight vector of a neural network The optimal value; express The partial derivatives; express The partial derivatives; S63: Designing an approximate optimal control law based on S62 for:

[0040] S64: Applying the Hamilton-Jacobi-Bellman equation In Replace with ,Will Replace with ,Will Replace with This yields the approximate Hamiltonian function. for:

[0041] S65: Based on Approximate Optimal Control Law Combining the approximate Hamiltonian function, the simplified approximate Hamiltonian function is obtained as follows:

[0042] S66: Define the Hamiltonian function error based on the simplified approximate Hamiltonian function. for:

[0043] The weight update rule of the neural network is designed based on the Hamiltonian function error. for:

[0044] In the formula: Indicates intermediate parameters and ; Indicates learning rate and ; Indicates intermediate parameters and ; Furthermore, the robust optimal controller constructed in S7 is: .

[0045] This invention provides a robust optimal control method for maintaining the state constraint structure of a mobile robotic arm, with the following beneficial effects: (1) The present invention maps the time-varying state constraint to the coordinate transformation of the unconstrained space by introducing a barrier function, thereby ensuring that the original state is always within the preset time-varying boundary interval; (2) The present invention integrates the designed linear extended state observer, whose gain is generated by the radial basis neural network and is adaptively tuned online through the adaptive moment estimator optimizer to estimate the lumped transformation disturbance, thereby achieving active disturbance rejection while maintaining the energy dissipation structure and closed-loop stability; (3) Construct a performance index function based on the transformed position error and the transformed offset Hamiltonian; design the optimal value function, i.e., the cost function; this invention can be seamlessly integrated with constraint processing based on obstacle function, ensuring optimality and strict constraint satisfaction while avoiding the computational burden of iterative learning, and the resulting controller only requires a small number of online learning parameters; the cost function is entirely composed of the tracking error and offset Hamiltonian within the port Hamiltonian framework after the obstacle function transformation. Unlike traditional optimal methods that require zero balance assumptions, discount factors, or iterative solutions to the Hamilton-Jacobi-Bellman equations, the proposed cost function remains naturally finite when the error converges, reducing design complexity and enhancing engineering practicality; (4) The present invention constructs a robust optimal controller, that is, designs a composite controller that combines feedforward compensation and feedback optimization based on offset Hamiltonian to ensure constraint satisfaction and asymptotic tracking control of trajectory. Attached Figure Description

[0046] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0047] Figure 1 This is a flowchart of the robust optimal control method for maintaining the state constraint structure of the mobile robotic arm according to the present invention; Figure 2 This is a schematic diagram of the control framework of the mobile robotic arm in this embodiment; Figure 3 This is a simulation diagram of the trajectory tracking in the XY plane using the controller proposed in this embodiment; Figure 4 This is a simulation diagram of the tracking error for all four degrees of freedom in this embodiment; Figure 5 This is a simulation diagram of the state constraint effect in the X direction in this embodiment; Figure 6 This is a simulation diagram of the state constraint effect in the Y direction in this embodiment; Figure 7 This is a simulation diagram of the communication topology in this embodiment; Figure 8 This is a simulation diagram of generalized momentum tracking under constraints in this embodiment; Figure 9 This is a simulation diagram of the evolution of the critic neural network weights in this embodiment. Detailed Implementation

[0048] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0049] This embodiment provides a robust optimal control method for a mobile robotic arm that maintains state constraints. By establishing a port Hamiltonian model of the mobile robotic arm, the system constraints and control objectives are clearly defined. The dynamic equations of the mobile robotic arm with nonholonomic constraints are constructed, and by defining the system state vector and generalized momentum, the original system dynamics and error dynamics are uniformly expressed in standard port Hamiltonian form. Simultaneously, basic assumptions such as bounded external disturbances and continuous reference trajectory are clarified, and three major control objectives are determined: ensuring state constraints, achieving trajectory tracking, and designing an optimal reinforcement learning controller. Figure 1 As shown, it includes: S1: Establish a port Hamiltonian dynamics model for the mobile robotic arm; and the port Hamiltonian dynamics model includes unknown transformation perturbation terms, specifically including the following steps: S11: The dynamic model of the mobile robotic arm subject to nonholonomic constraints is established as follows:

[0050]

[0051] In the formula: Represents the generalized inertia matrix; Represents the matrix of Coriolis force and centrifugal force; Represents the gravity vector; This represents the active control input torque vector; Represents a nonholonomic constraint matrix; Represents the constraint force vector; Denotes the lumped disturbance vector and ; Represents the generalized velocity vector of the system first derivative and ; Indicates transpose; This indicates the velocity of the mobile platform along the X and Y axes in the geodetic coordinate system; This indicates the yaw rate of the preset mobile platform; This represents the joint angular velocity of the robotic arm links in a preset geodetic coordinate system, namely the joint angular velocity of the first link and the joint angular velocity of the second link of the robotic arm. Represents independent velocity vectors first derivative and ; These represent the angular velocities of the left drive wheel and the right drive wheel of the mobile platform, respectively. This represents the joint angular velocity of the first and second links of the robotic arm in a preset reduced-order coordinate system; Represents the uncertainty term in the model; This indicates external interference; Represents the intermediate parameter matrix; The null space matrix represents the nonholonomic constraint matrix A, used to represent the independent velocity vectors of the system. Mapped to the system's generalized velocity vector In this embodiment, S spans the null space of A, meaning that the column vectors of S form a basis for the null space of A, such that the matrix multiplication relation... This holds true, thus enabling the transformation of constrained system dynamics into an equivalent unconstrained reduced-order system while satisfying the system's nonholonomic constraints. S12: The reduced-order model obtained from the mobile robotic arm dynamics model is:

[0052]

[0053] In the formula: express The first derivative; Indicates intermediate parameters; express The first derivative; S13: Define the system state vector of the mobile robotic arm according to the reduced-order model. for:

[0054]

[0055] In the formula: Represents the position coordinates of the mobile robotic arm first derivative and ; Indicates the location coordinates of the mobile platform; This indicates the position coordinates of the end effector of the mobile robotic arm; Represents the kinematic Jacobian matrix of the system; This represents generalized momentum; in this embodiment, to represent the system in port Hamiltonian form, the state vector is defined as... ; S14: The Hamiltonian dynamics model of the mobile robotic arm, based on S13, is as follows:

[0056]

[0057]

[0058] In the formula: Representing the Hamiltonian function Regarding the system state vector gradient vector and ; This represents the potential energy of the system. Since we only consider the motion in the XY plane, the potential energy can be neglected, i.e. ; This represents the antisymmetric commutative term in the Coriolis matrix after order reduction; This represents the damping coefficient components in the damping matrix; Represents the system state vector The input matrix and ; Represents the identity matrix; Represents the system state vector The parameter matrix; Represents the system state vector Unknown transformation perturbation term and In this embodiment If it is partially symmetric, then we can define... ,in These represent the antisymmetric commutation terms in the equivalent generalized inertia matrix and the Coriolis matrix after the reduced-order transformation, respectively. In this embodiment, by utilizing the oblique symmetry property, the original system dynamics and error dynamics can be represented in standard port Hamiltonian form, respectively. S2: Construct the desired motion equation based on the port Hamiltonian dynamics model; design a feedforward controller based on the desired motion equation; define the offset Hamiltonian quantity based on the port Hamiltonian dynamics model and the desired motion equation, specifically including the following steps: S21: Given the desired trajectory and Based on the Hamiltonian dynamics model of the port, the desired equation of motion is constructed as follows:

[0059] In the formula: express The first derivative; express The abbreviated form; Represents the desired position coordinates and desired generalized momentum of the moving robotic arm; Represents a given desired trajectory The parameter matrix; Represents a given desired trajectory The unknown transformation disturbance term; This represents a feedforward controller; in this embodiment, a feedforward controller is first designed at the physical layer. To ensure that the system can track the desired trajectory without interference and uncertainty, then... Substituting into the port Hamiltonian dynamics, we obtain the desired equation of motion; Represents a given desired trajectory The input matrix; where This represents the unknown perturbation in the desired state. In this embodiment, the perturbation of the feedforward design is ignored. Setting the term to zero yields the algebraic condition: Since the input matrix is Its Moore-Penrose pseudo-inverse is ; S22: Based on algebraic conditions, the feedforward controller is designed according to the desired motion equation as follows:

[0060] In the formula: express The first derivative; This represents the kinematic Jacobian matrix corresponding to the desired equation of motion; Let and represent the partial derivatives of the Hamiltonian with respect to the position and momentum components, respectively, and all terms are in the desired state. Evaluation below; express The abbreviated form; express The abbreviated form; in this embodiment From expected acceleration Through relational formulas get; S23: Based on the port Hamiltonian dynamics model and the desired motion equation, the offset Hamiltonian is defined as:

[0061] In the formula: This represents the offset Hamiltonian. In this embodiment, the displacement Hamiltonian has the following properties: ,when exist In the event of a severe emergency, exist Within the neighborhood of It is positive definite; in this embodiment, based on the ideal reference trajectory of the mobile robotic arm, the energy structure of the physical layer is designed to maintain the feedforward controller; based on the input matrix structure of the port Hamiltonian system, the feedforward control law is derived analytically through algebraic conditions while ignoring disturbances, to ensure that the system can accurately track the desired trajectory in the ideal state; at the same time, the shifted Hamiltonian function is defined and its positive definiteness in the neighborhood of the desired trajectory is proved, providing a physical energy index for the subsequent design of the optimal feedback controller; S3: Based on the port Hamiltonian dynamics model, a coordinate transformation strategy is constructed by introducing a barrier function to map the time-varying state constraints of the robotic arm to an unconstrained space; the offset Hamiltonian is converted into a transformed offset Hamiltonian through the coordinate transformation strategy, specifically including the following steps: S31: In this embodiment, in order to maintain the original state Time-varying state constraints are implemented, and a general barrier function is applied to each constrained component. To implement these constraints, the original state... Each constrained component is assigned a general barrier function, and the transformed unconstrained state is defined as follows: and The time-varying constraints are defined based on the port Hamiltonian dynamics model. ; These represent the upper and lower bounds of the time-varying constraint, respectively. ; Represents the original state The Each of the constrained state components Both are achieved through mapping from the corresponding constraint variables. The obtained, and transformed, unconstrained state The formula for obtaining it is:

[0062]

[0063] In the formula: express The abbreviated form; Indicate design parameters and ; Represents the barrier function; in this embodiment, this transformation guarantees that as long as It is finite. Strictly maintain Inside; S32: Based on S31, the transformed unconstrained state derivative Represented as:

[0064]

[0065]

[0066]

[0067] In the formula: express The abbreviated form is defined by the partial derivative of the barrier function; express The Middle One element; express The Middle One element; express The abbreviated form; Represents the parameter matrix; express The first derivative; express The abbreviated form; S33: Define the desired transformation state as and express The abbreviated form; based on S32 combined with the desired transformation state is Obtain the transformed tracking error and ; In this embodiment, by using... This represents the desired transformed state, and the tracking error after the transformation. , Let represent the unconstrained state vector at the current moment after being mapped by a unified obstacle function, and the transformed tracking error expression is:

[0068] according to The definition can be obtained Due to the transformation Typically time-varying and nonlinear, the desired trajectory in the transformed coordinate system. It does not have a port Hamiltonian representation; however, its time derivative can be obtained by applying... Differentiating over time and substituting into the expected dynamics (omitting the perturbation term) yields:

[0069] In the formula: , , In this embodiment, a reversible potential field transformation is given. and reverse The displacement Hamiltonian in the transformed coordinate system can be defined as the transformed tracking error. and To obtain the scalar energy function for:

[0070] in It represents the shift Hamiltonian with respect to the tracking error state in the transformed unconstrained coordinate system; This represents the desired trajectory mapped to the unconstrained space; function That is, scalar energy function From the offset Hamiltonian It inherited the following key properties:

[0071]

[0072] From this, we can conclude that: if exist The location is strictly convex and has an inverse mapping. If it is smooth, then exist Within the neighborhood of It is positive definite; for practical controller synthesis or stability analysis, it can be adopted... exist The local quadratic approximation in the vicinity is expressed as:

[0073] In the formula: the Hessian matrix is ​​obtained by using the chain rule from... The Jacobian matrix is ​​calculated from the inverse transformation. This quadratic form is consistent with standard Lyapunov analysis, facilitating the design of optimal feedback controllers under state constraints. In this embodiment, feedforward conditions are used to cancel the condition. The estimated term can yield a compact derivative of the tracking error; S34: Based on the definition in S33, obtain the derivative of the compact tracking error. for:

[0074] In the formula: This represents the tracking error vector of the system position components after obstacle function transformation; This represents the tracking error vector of the system's generalized momentum components after the barrier function transformation; This represents the drift term or nominal system dynamics term in the transformed error dynamics equation; Indicates the transformed disturbance and ;in satisfy Furthermore, it is locally Lipschitz near the origin; This represents the optimal feedback control input. S35: In this embodiment, to facilitate optimal control design under state constraints, the shifted Hamiltonian described in the previous formula is used. As a Lyapunov candidate function, its time derivative along the error dynamics is: Note in this implementation and Inheriting the port Hamiltonian structure, it is possible to derive stable and optimal feedback controllers by utilizing partial symmetry and damping characteristics. ; The coordinate transformation strategy described above converts the offset Hamiltonian into a transformed offset Hamiltonian, that is, it converts the offset Hamiltonian into a transformed offset Hamiltonian based on S34. That is, by using Taylor series expansion and following linearization proof, the transformed displacement Hamiltonian... It can be locally approximated by a quadratic form as:

[0075] In the formula: Represents the Hessian matrix; The quadratic energy weight matrix represents the generalized momentum components in the transformed error space; where the Hessian matrix is... It is given by the following formula: Therefore, its Hessian matrix in the desired state is: In this embodiment, due to It is a diagonal matrix and can be divided into ; Let each represent the unified obstacle function Jacobian matrix at the desired trajectory. For the diagonal block matrix of the position component and the generalized momentum component, we can obtain The absence of the position error term in this embodiment reflects the energy landscape relative to the mean, ignoring potential energy. It is flat. To enforce position tracking, the objective function of the optimal control design will be directly introduced with respect to... The explicit quadratic penalty term is a common practice in trajectory tracking problems. This method maintains consistency with the displacement Hamiltonian framework while also taking into account the specific characteristics of the system. In this embodiment, a general obstacle function is used to map the constraint state, transforming the constrained original dynamic system into an unconstrained error dynamic system. For each constrained component of the original state, a novel obstacle mapping based on the hyperbolic tangent function is constructed, its time derivative is derived, and an explicit correspondence between the unconstrained error state and the original constrained state is established. Based on the chain rule of inverse transformation, the displacement Hamiltonian function is locally approximated in the transformed coordinate system, and finally the constrained system is compactly expressed as an unconstrained error dynamic equation.

[0076] S4: Design a linear extended state observer, and estimate the unknown transformation disturbance term using the linear extended state observer to obtain the estimated disturbance. Specific steps include: S41: To compensate for unknown transformation disturbances in real time In this embodiment, a linearly extended state observer is designed to... Each component is treated as an extended state for estimation; the gain of the linear extended state observer is adjusted online by a compact radial basis function neural network, which is trained through an adaptive moment estimator optimizer, thus enabling it to adapt to changing perturbation patterns without manual retuning. Its detailed block diagram is shown below. Figure 2 As shown. In this embodiment, for each momentum channel ,definition , Indicates the first The tracking error of a generalized momentum channel after the unified barrier function transformation, and let the extended state This represents the lumped disturbance acting on the channel; where The dynamic equation is ,here express The One component; and Based on empirical values ​​calculated from the current state and feedback control, a second-order linear extended state observer is constructed for this channel, with the following expression:

[0077] In the formula: express The One component; and Indicates the first The tracking error of a generalized momentum channel after barrier function transformation; Indicates extended state and Indicates the action in the Lumped disturbances on the channel; express The estimated value; express The first derivative; express The abbreviated form; The value represents the observer gain coefficient generated online adaptively by the RBF neural network; during the actual observation operation, it is determined by the network output value. and Directly used as the feedback gain coefficient in the observer equation and The estimation calculation of participating states and disturbances is performed using... To each and To make an estimate; therefore, Provided the first One interference component Online estimation; S42: Estimate the unknown transformation disturbance term using the linear extended state observer to obtain the estimated disturbance. and In this embodiment, the estimated disturbance is used... It is incorporated into subsequent control laws to achieve active suppression of disturbances.

[0078] In this embodiment, the observer gain of the linear extended state observer and This has a crucial impact on estimation accuracy and convergence speed. In this embodiment, to avoid manual parameter tuning and ensure adaptability, each pair of gains is generated by a small radial basis function neural network with three structures; the input vector... Due to tracking error Its derivative and expected value Composition; hidden layers use Gaussian activation function , ,in and These represent the center and width, respectively. The output is a linear combination: All network parameters Online updates to minimize instantaneous estimation error In this embodiment, an adaptive moment estimation optimizer is used instead of the classic gradient descent with a fixed learning rate. This optimizer maintains an adaptive learning rate through moment estimation of the gradient; for a general parameter... ,set up Indicate its gradient; first moment estimate Second Moment Estimation Updated to:

[0079]

[0080] Where: decay rate ;set up Indicates the iteration step size (starting from 1); the moment estimate is initialized to... , The bias-corrected estimate is given by the following formula:

[0081] The general parameters The update formula is: In the formula: and The parameters are positive. In this embodiment, the method significantly accelerates convergence and stabilizes the learning process, enabling the observer gain to quickly adapt to constantly changing disturbance conditions. In this embodiment, a linear extended state observer based on radial basis function neural network and adaptive moment estimation is designed to achieve online adaptive adjustment of the disturbance estimation gain. The lumped disturbance in the error dynamics is regarded as an extended state to construct a second-order linear extended state observer, and the observer gain parameters are generated in real time using a compact radial basis function neural network. At the same time, an adaptive moment estimation optimizer is introduced to update the network parameters online, avoiding manual parameter tuning, so as to significantly improve the speed and accuracy of the observer's adaptive estimation of time-varying disturbances. S5: Construct a performance index function based on the transformed offset Hamiltonian; obtain the optimal value function based on the Bellman optimality principle using the performance index function; construct the Hamilton-Jacobi-Bellman equation based on the optimal value function, specifically including the following steps: S51: The performance index function is constructed based on the transformed position error and the transformed offset Hamiltonian as follows:

[0082] In the formula: Represents a performance metric function; This represents the discount factor, ensuring that the integral converges; The energy weight matrix represents the quadratic form of the position components in the transformed error space, and... , Indicates the weighting factor; This represents the diagonal matrix corresponding to the position in the desired trajectory. In this embodiment, a zero-sum differential game is constructed based on the transformed error dynamics, where the controller minimizes a cost functional, while the disturbance maximizes the cost functional. To make the problem well-posed, this embodiment adds a quadratic penalty term for the control input and the disturbance to the cost, thereby obtaining the performance index function. S52: Based on the Bellman optimality principle, the optimal value function is obtained from the performance index function as follows:

[0083] In the formula: Represents the optimal value function; Indicates a time constant; Indicates the discount factor; Represents the optimal value function Regarding the error state after transformation The gradient; Indicates the nominal dynamic item The generalized momentum component; express The optimal value function corresponding to the time step; This indicates that after the barrier function transformation, The tracking error vector of the system position component corresponding to the time step; in this embodiment, the optimal value function can be defined as ; express The partial derivatives and ; S53: Construct the Hamilton-Jacobi-Bellman equation based on the optimal value function. for:

[0084] or

[0085] In the formula: Represents the optimal value function to be solved; Indicates the nominal dynamic item The generalized momentum component; in this embodiment, based on the transformed unconstrained error dynamics, a zero-sum differential game framework is constructed, and the optimal feedback control strategy is derived; a zero-sum differential game problem of controller minimization and disturbance maximization is established on the transformed error system, and a performance index function composed of shift Hamiltonian and position tracking penalty term is designed; the Hamilton-Jacobi-Bellman equation is derived based on the Bellman optimality principle, and the optimal feedback control law expressed by the gradient of the optimal value function is obtained by minimization solution, and the optimal solution is approximated by the strategy iteration algorithm; S6: A neural network is used to approximate the optimal value function to design an approximate optimal control law; simultaneously, a simplified approximate Hamiltonian function is obtained based on the approximate optimal value function and the Milton-Jacobi-Bellman equation; the Hamiltonian function error is defined according to the simplified approximate Hamiltonian function; the weight update law of the neural network is designed according to the Hamiltonian function error; the approximate optimal control law is updated according to the weight update law to obtain the optimal feedback control law, specifically including the following steps: S61: Solve the Hamilton-Jacobi-Bellman equations based on the Bellman optimality principle. And obtain information about Optimal strategy for:

[0086] In the formula: Indicates intermediate parameters and In order to solve This embodiment employs a policy iteration algorithm that alternates between policy evaluation and policy improvement, specifically including: To numerically implement the policy iteration algorithm, this embodiment approximates... It is a neural network that utilizes physical energy structures, and its expression is:

[0087] In the formula: It is a neural network that approximates the true optimal value function. The residuals generated during the process; It is a vector of linearly independent basis functions; It is the weight vector that needs to be optimized; S62: Using neural networks to approximate the optimal value function to rewrite the optimal strategy for:

[0088] In the formula: This represents the function by which a neural network approximates the true optimal value. The residuals generated during the process; A vector representing linearly independent basis functions; Represents the weight vector of a neural network The optimal value; express The partial derivatives; express The partial derivatives; S63: In this embodiment, the approximate performance index function is expressed as: Then, based on S62, an approximate optimal control law can be designed. for:

[0089] S64: Applying the Hamilton-Jacobi-Bellman equation In Replace with ,Will Replace with ,Will Replace with This yields the approximate Hamiltonian function. for:

[0090] S65: Based on Approximate Optimal Control Law Combining the approximate Hamiltonian function, the simplified approximate Hamiltonian function is obtained as follows:

[0091] S66: Define the Hamiltonian function error based on the simplified approximate Hamiltonian function. for:

[0092] The weight update rule of the neural network is designed based on the Hamiltonian function error. for:

[0093] In the formula: Indicates intermediate parameters and ; ; Indicates learning rate and In this embodiment, after the critic network weights converge, the value function is obtained. and its gradient The approximation is obtained by using a critical neural network to approximate the optimal value function, designing a weight update law and achieving closed-loop control stability. A critical neural network with a physical energy structure is constructed to approximate the optimal value function, and an adaptive update law for weights is designed based on minimizing Hamiltonian error. This is used for subsequent integrated feedforward control, neural network-based feedback optimal control, and observer disturbance compensation to construct the overall control law, i.e., the robust optimal controller.

[0094] S7: Based on the optimal feedback control law, estimated disturbance, and feedforward controller, construct a robust optimal controller; the robust optimal controller constructed in this embodiment is:

[0095] In this embodiment, the estimated disturbance is used... By incorporating it into the control law, active suppression of disturbances can be achieved.

[0096] It is the feedforward term, i.e., the feedforward controller. The optimal feedback, or optimal feedback control law, is derived from the shifted Hamiltonian. By directly compensating for disturbances, the closed-loop system restores the nominal performance of the optimal controller while ensuring that all signals remain bounded. The robust optimal controller described above enables a structure-preserving robust optimal control process for a mobile robotic arm oriented towards state constraints.

[0097] The method described in this embodiment transforms a complex time-varying state-constrained system into an equivalent unconstrained system by introducing a general obstacle function. Based on the shifted Hamiltonian function within the port Hamiltonian framework and differential game theory, a composite control strategy incorporating feedforward compensation and feedback optimization is constructed. Furthermore, the method integrates a linear extended state observer based on radial basis function neural networks and adaptive moment estimation, achieving high-precision estimation and active disturbance rejection of lumped disturbances, offering the following advantages: (1) A general obstacle function was designed to systematically embed time-varying output constraints into the unconstrained port Hamiltonian representation; using this transformation, a composite controller combining feedforward compensation and displacement Hamiltonian-based feedback optimization was designed to ensure constraint satisfaction and trajectory asymptotic tracking.

[0098] (2) A linear extended state observer of the design is integrated, the gain of which is generated by a radial basis function neural network and is adaptively tuned online by an adaptive moment estimator optimizer to estimate lumped transformation perturbations; thereby achieving active disturbance suppression while maintaining the energy dissipation structure and closed-loop stability. (3) Establish the optimal value function, i.e., the cost function. The cost function is constructed only by the tracking error and the displacement Hamiltonian within the port Hamiltonian framework of the general barrier function transformation. Unlike the traditional optimal method, which requires the zero balance assumption, discount factor or iterative Hamilton-Jacobi-Bellman solution, the proposed cost function naturally remains finite when the error converges, thereby reducing the design complexity and enhancing practical applicability. At the same time, in this embodiment, the system with complex constraints such as nonholonomic constraints and time-varying output constraints is transformed into an unconstrained error dynamic model through the designed general barrier function mapping. Based on this, the inherent interconnection and damping matrix structure of the port Hamiltonian system is utilized. Through the local quadratic approximation of the shifted Hamiltonian function in the error coordinate system and combined with differential game theory, the optimal feedback control law that preserves the structure is designed. Finally, the feedforward control, optimal feedback and disturbance rejection observer are integrated to ensure that the energy dissipation characteristics of the constrained nonlinear system are not destroyed. All state signals can strictly satisfy the time-varying boundary constraints and achieve asymptotic convergence of the closed-loop error.

[0099] The system framework of the state-constrained, structure-preserving robust optimal control method for a mobile robotic arm described in this embodiment is as follows: Figure 2 As shown. Figure 2 The control structure of the method described in this embodiment is illustrated, dividing the entire control process into three collaborative modules: the first part is the state constraint mapping layer, which establishes a bidirectional mapping between the original state and the unconstrained error space based on a unified obstacle function, and introduces a shifted Hamiltonian as an energy index; the second part is the optimal feedback control strategy, which constructs a cost function based on a zero-sum differential game, iteratively approximates the optimal value function through a critical neural network, and outputs the optimal feedback control law; the third part is the disturbance rejection compensation module, which uses an adaptive moment estimation-optimized radial basis neural network adaptive observer to estimate and compensate for the lumped disturbances suffered by the system in real time.

[0100] To verify the effectiveness of the method described in this embodiment, a simulation model of a two-link mobile manipulator was built using the MATLAB / Simulink environment. In this simulation experiment, the method described in this embodiment pre-determined a four-degree-of-freedom mobile manipulator system, commanding its mobile platform and end effector to track a specified time-varying trajectory, and applied strict position and momentum boundary constraints. Figure 3 The actual trajectory tracking curves of the mobile platform and the end effector in the XY plane are shown (in the figure, Ideal(MP) is the ideal trajectory of the mobile platform, Actual(MP) is the actual trajectory of the mobile platform, Ideal(EE) is the ideal trajectory of the end effector, Actual(EE) is the actual trajectory of the end effector, Link1 is the first link of the robotic arm, and Link2 is the second link of the robotic arm;). It can be seen that the actual trajectory and the expected trajectory are highly coincident. Figure 4 The convergence process of the four tracking error components is shown. Figures 5 to 8 The system's state constraint effect in the X and Y directions and generalized momentum is demonstrated. All trajectories strictly do not cross the preset time-varying boundary, which fully verifies the constraint guarantee capability of the method described in this embodiment. Figure 9 The evolution of the weights in the critique neural network is demonstrated. Quantitative comparison results show that, compared with traditional methods such as PH-TCC and LG-TCC, the method described in this embodiment significantly improves the performance metrics of IAE, ISE, and TIAE, proving its superiority and robustness.

[0101] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A robust optimal control method for maintaining the state constraints of a mobile robotic arm, characterized in that, Specifically, the following steps are included: S1: Establish a port Hamiltonian dynamics model for the mobile robotic arm; and the port Hamiltonian dynamics model includes unknown transformation perturbation terms; S2: Construct the desired motion equation based on the port Hamiltonian dynamics model; design a feedforward controller according to the desired motion equation; define the offset Hamiltonian amount according to the port Hamiltonian dynamics model and the desired motion equation; S3: Based on the port Hamiltonian dynamics model, a coordinate transformation strategy is constructed by introducing a barrier function to map the time-varying state constraints of the robotic arm to an unconstrained space; the offset Hamiltonian is converted into a transformed offset Hamiltonian through the coordinate transformation strategy. S4: Design a linear extended state observer, estimate the unknown transformation disturbance term based on the linear extended state observer, and obtain the estimated disturbance; S5: Construct a performance index function based on the transformed offset Hamiltonian; obtain the optimal value function based on the performance index function according to the Bellman optimality principle; construct the Hamilton-Jacobi-Bellman equation based on the optimal value function; S6: A neural network is used to approximate the optimal value function to design an approximate optimal control law; at the same time, a simplified approximate Hamiltonian function is obtained by combining the approximate optimal value function with the Milton-Jacobi-Bellman equation. The Hamiltonian function error is defined based on the simplified approximate Hamiltonian function; the weight update rule of the neural network is designed based on the Hamiltonian function error; the approximate optimal control law is updated based on the weight update rule, and the optimal feedback control law is obtained. S7: Construct a robust optimal controller based on the optimal feedback control law, estimated disturbance, and feedforward controller; implement the structure-maintaining robust optimal control process of the mobile robotic arm oriented towards state constraints based on the robust optimal controller.

2. The robust optimal control method for maintaining the state constraint structure of a mobile robotic arm according to claim 1, characterized in that, The method for establishing the port Hamiltonian dynamics model of the mobile robotic arm in S1 specifically includes the following steps: S11: The dynamic model of the mobile robotic arm subject to nonholonomic constraints is established as follows: In the formula: Represents the generalized inertia matrix; Represents the matrix of Coriolis force and centrifugal force; Represents the gravity vector; This represents the active control input torque vector; Represents a nonholonomic constraint matrix; Represents the constraint force vector; Denotes the lumped disturbance vector and ; Represents the generalized velocity vector of the system first derivative and ; Indicates transpose; This indicates the velocity of the mobile platform along the X and Y axes in the geodetic coordinate system; This indicates the yaw rate of the preset mobile platform; This represents the joint angular velocity of the robotic arm link in a preset geodetic coordinate system; Represents independent velocity vectors first derivative and ; These represent the angular velocities of the left drive wheel and the right drive wheel of the mobile platform, respectively. This represents the joint angular velocity of the robotic arm link in a preset reduced-order coordinate system; Represents the uncertainty term in the model; This indicates external interference; Represents the intermediate parameter matrix; Let A be the null space matrix representing the nonholonomic constraint matrix A; S12: The reduced-order model obtained from the mobile robotic arm dynamics model is: In the formula: express The first derivative; Indicates intermediate parameters; express The first derivative; S13: Define the system state vector of the mobile robotic arm according to the reduced-order model. for: In the formula: Represents the position coordinates of the mobile robotic arm first derivative and ; Indicates the location coordinates of the mobile platform; This indicates the position coordinates of the end effector of the mobile robotic arm; Represents the kinematic Jacobian matrix of the system; Represents generalized momentum; S14: The Hamiltonian dynamics model of the mobile robotic arm, based on S13, is as follows: In the formula: Representing the Hamiltonian function Regarding the system state vector gradient vector and ; This represents the potential energy of the system; This represents the antisymmetric commutative term in the Coriolis matrix after order reduction; This represents the damping coefficient components in the damping matrix; Represents the system state vector The input matrix and ; Represents the identity matrix; Represents the system state vector The parameter matrix; Represents the system state vector Unknown transformation perturbation term and .

3. The robust optimal control method for maintaining the state constraint structure of a mobile robotic arm according to claim 2, characterized in that, S2 specifically includes the following steps: S21: Given the desired trajectory and Based on the Hamiltonian dynamics model of the port, the desired equation of motion is constructed as follows: In the formula: express The first derivative; express The abbreviated form; Represents the desired position coordinates and desired generalized momentum of the moving robotic arm; Represents a given desired trajectory The input matrix; Represents a given desired trajectory The parameter matrix; Represents a given desired trajectory The unknown transformation disturbance term; Indicates a feedforward controller; S22: The feedforward controller is designed based on the desired motion equation as follows: In the formula: express The first derivative; This represents the kinematic Jacobian matrix corresponding to the desired equation of motion; Let represent the partial derivatives of the Hamiltonian with respect to the position and momentum components, respectively; express The abbreviated form; express The abbreviated form; S23: Based on the port Hamiltonian dynamics model and the desired motion equation, the offset Hamiltonian is defined as: In the formula: This represents the offset Hamiltonian.

4. The robust optimal control method for maintaining the state constraint structure of a mobile robotic arm according to claim 3, characterized in that, S3 specifically includes the following steps: S31: Define the unconstrained state after the transformation as... and ; Based on the port Hamiltonian dynamics model, the time-varying constraints are defined as follows: ; These represent the upper and lower bounds of the time-varying constraint, respectively. Represents the original state The A constrained state component; And the transformed unconstrained state The formula for obtaining it is: In the formula: express The abbreviated form; Indicate design parameters and ; Represents the barrier function; S32: Based on S31, the transformed unconstrained state derivative Represented as: In the formula: express The abbreviated form; express The Middle One element; express The Middle One element; express The abbreviated form; Represents the parameter matrix; express The first derivative; express The abbreviated form; S33: Define the desired transformation state as and express The abbreviated form; based on S32 combined with the desired transformation state is Obtain the transformed tracking error and ; S34: Based on the definition in S33, obtain the derivative of the compact tracking error. for: In the formula: This represents the tracking error vector of the system position components after obstacle function transformation; This represents the tracking error vector of the system's generalized momentum components after the barrier function transformation; This represents the drift term or nominal system dynamics term in the transformed error dynamics equation; Indicates the transformed disturbance and ; This represents the optimal feedback control input. S35: Convert the offset Hamiltonian into a transformed offset Hamiltonian using the coordinate transformation strategy, i.e., convert the offset Hamiltonian into a transformed offset Hamiltonian based on S34. for: In the formula: Represents the Hessian matrix; The quadratic energy weight matrix represents the generalized momentum components in the transformed error space.

5. The robust optimal control method for maintaining the state constraint structure of a mobile robotic arm according to claim 4, characterized in that, S4 specifically includes the following steps: S41: Design the linear extended state observer as follows: In the formula: express The One component; and Indicates the first The tracking error of a generalized momentum channel after barrier function transformation; Indicates extended state and Indicates the action in the Lumped disturbances on the channel; express The estimated value; express The first derivative; express The abbreviated form; Denotes the observer gain coefficients adaptively generated online by the RBF neural network; S42: Estimate the unknown transformation disturbance term using the linear extended state observer to obtain the estimated disturbance. and .

6. The robust optimal control method for maintaining the state constraint structure of a mobile robotic arm according to claim 5, characterized in that, S5 specifically includes the following steps: S51: The performance index function is constructed based on the transformed position error and the transformed offset Hamiltonian as follows: In the formula: Represents a performance metric function; Indicates the discount factor; The energy weight matrix represents the quadratic form of the position components in the transformed error space, and... , Indicates the weighting factor; This represents the diagonal matrix corresponding to the position in the desired trajectory; S52: Based on the Bellman optimality principle, the optimal value function is obtained from the performance index function as follows: In the formula: Represents the optimal value function; Indicates a time constant; Indicates the discount factor; Represents the optimal value function Regarding the error state after transformation The gradient; Indicates the nominal dynamic item The generalized momentum component; express The optimal value function corresponding to the time step; This indicates that after the barrier function transformation, The tracking error vector of the system position component corresponding to the time step; S53: Construct the Hamilton-Jacobi-Bellman equation based on the optimal value function. for: In the formula: Represents the optimal value function to be solved; Indicates the nominal dynamic item The generalized momentum component.

7. The robust optimal control method for maintaining the state constraint structure of a mobile robotic arm according to claim 6, characterized in that, S6 specifically includes the following steps: S61: Solve the Hamilton-Jacobi-Bellman equations based on the Bellman optimality principle. And obtain information about Optimal strategy for: In the formula: Indicates intermediate parameters and ; S62: Using neural networks to approximate the optimal value function to rewrite the optimal strategy for: In the formula: This represents the function by which a neural network approximates the true optimal value. The residuals generated during the process; A vector representing linearly independent basis functions; Represents the weight vector of a neural network The optimal value; express The partial derivatives; express The partial derivatives; S63: Designing an approximate optimal control law based on S62 for: S64: Applying the Hamilton-Jacobi-Bellman equation In Replace with ,Will Replace with ,Will Replace with This yields the approximate Hamiltonian function. for: S65: Based on Approximate Optimal Control Law Combining the approximate Hamiltonian function, the simplified approximate Hamiltonian function is obtained as follows: S66: Define the Hamiltonian function error based on the simplified approximate Hamiltonian function. for: The weight update rule of the neural network is designed based on the Hamiltonian function error. for: In the formula: Indicates intermediate parameters and ; Indicates learning rate and ; Indicates intermediate parameters and 。 8. The robust optimal control method for maintaining the state constraint structure of a mobile robotic arm according to claim 7, characterized in that, The robust optimal controller built in S7 is: 。

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