CN116690561A - Self-adaptive optimal backstepping control method and system for single-connecting-rod mechanical arm - Google Patents
Self-adaptive optimal backstepping control method and system for single-connecting-rod mechanical arm Download PDFInfo
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Abstract
The application discloses a self-adaptive optimal backstepping control method and a self-adaptive optimal backstepping control system for a single-link mechanical arm, which belong to the technical field of self-adaptive optimal backstepping control of the single-link mechanical arm and comprise the following steps: obtaining a dynamic model of the single-link robot mechanical arm, and converting the model into a state model by obtaining physical characteristics of the single-link robot mechanical arm; based on a state model, an optimal controller for the single-link robot mechanical arm is designed by establishing an expected track model of a reference signal and giving a time-varying partial position constraint condition of the joint angular displacement of the single-link robot and combining a backstepping technique and a reinforcement learning algorithm, so as to carry out self-adaptive optimal backstepping control on the single-link robot mechanical arm; according to the application, a system which needs to meet the position limiting requirement in the whole course is converted into a system which only needs to meet the limiting requirement at a specific moment by designing a conversion function and a special barrier function, so that the constraint condition of time-varying position limitation of the joint angular displacement part of the single-link mechanical arm system is realized.
Description
Technical Field
The application belongs to the technical field of self-adaptive optimal backstepping control of single-link mechanical arms, and particularly relates to a self-adaptive optimal backstepping control method and system for a single-link mechanical arm.
Background
The human science and technology is changed day by day, and robots capable of effectively reducing the burden and work of human beings are produced, and the continuous progress of human society is further promoted by the production and development of the robots. The mechanical arm is used as the most widely used executing device in the robot, has been developed rapidly in recent years, has been widely applied to different fields of actual life by virtue of unique advantages of the mechanical arm, and can be designed into different mechanical arm appearances and movement tracks according to different task requirements and precision requirements from processing of precision parts, minimally invasive surgery, production line tasks, military manufacturing and space laboratories, satellites, spacecrafts and the like. The flexible joint robot can operate in harmful or dangerous environments, so that the life safety of operators is protected, repetitive work tasks can be replaced, unnecessary labor capacity of the human is reduced, automatic operation is realized, the production efficiency is greatly improved, and the flexible joint robot has the advantages of small size, light weight, low consumption, high efficiency, flexibility, convenience and the like, and becomes an important field for deep research of people in the present and future.
In practical production, with the development of control technology and production requirements, the mechanical arm system is further affected by various uncertainty factors, which puts higher and higher demands on the control performance of the mechanical arm. For example, in operations such as space docking and medical surgery, high requirements are placed on response speed, anti-interference performance and steady-state control accuracy of the mechanical arm. Meanwhile, for certain specific work tasks, in consideration of the actual situation, the joint robot with the standard tracking performance is required to control the mechanical arm to effectively track the expected track, and also consider the time-varying partial position limited condition that the output of the control system meets the requirement within the specified limit, so that the mechanical arm is ensured to complete the task under the time-varying partial position limited condition. The control problem is not same as the control problem with high precision, the control problem under the condition that the position of the time-varying part is limited, the control method has clear limiting requirements on each performance index, and the mechanical arm needs to be controlled to complete the set task on the premise of meeting the limiting conditions. The control problem under the condition that the time-varying part position is limited is researched, on one hand, the problem that a mechanical arm system with the time-varying part position limited requirement is difficult to design a controller is solved, and on the other hand, the purpose of high-precision control of the mechanical arm can be achieved through an adaptive control method. Therefore, the control problem of the mechanical arm is researched, including control research and reinforcement learning ability research under the condition that the position of the time-varying part is limited, and the mechanical arm has important theoretical significance and practical value. And the mechanical arm control meeting the time-varying partial position constraint is researched, so that the control performance of the system is improved, and a theoretical basis is provided for the mechanical arm control under the appointed constraint environment.
Disclosure of Invention
In order to solve the problems, the application aims to provide a self-adaptive optimal backstepping control method of a time-varying part position limited single-link mechanical arm based on reinforcement learning, which is used for solving the problem that the joint angular displacement of a mechanical arm system is limited in time-varying part position.
In order to achieve the above technical object, the present application provides an adaptive optimal backstepping control method for a single-link mechanical arm, including:
obtaining a dynamic model of the single-link robot mechanical arm, and converting the model into a state model by obtaining physical characteristics of the single-link robot mechanical arm;
based on the state model, an optimal controller for the single-link robot mechanical arm is designed by establishing an expected track model of a reference signal and giving a time-varying partial position constraint condition of the joint angular displacement of the single-link robot and combining a backstepping technique and a reinforcement learning algorithm, so as to carry out self-adaptive optimal backstepping control on the single-link robot mechanical arm.
Preferably, in response to the establishment of the desired trajectory model, the desired trajectory model is expressed as:
y r ＝0.1(tanh(t-T s )+tanh(t-T r ))sin(t)+0.1cos(t)
wherein T represents time, T s Indicating the moment of onset of constrained joint angular displacement g, T r Indicating the end time at which the joint angular displacement g is constrained.
Preferably, in response to the process of obtaining time-varying part position constraints, the robotic arm system outputs according to a given single linkg and reference trajectory y r Generating time-varying partial position constraints, wherein the constraints are:
(1) When t is E [ t ] 0 ,T s ) When the joint angular displacement is not restrained;
(2) When T is E [ T ] s ,T r ) When the joint angular displacement is constrained, and k is satisfied l ＜g＜k u ；
(3) When T is E [ T ] r Infinity), the angular displacement of the joint is not constrained;
wherein t is 0 Representing the starting instant, k, of the system operation u And k l Representing the constrained upper and lower time-varying functions of the joint angular displacement g, respectively.
Preferably, in response to a design process of the optimal controller, the optimal virtual controller and the optimal actual controller are respectively designed based on a system dynamic equation of the dynamic model, and the optimal controller is constructed, wherein the optimal controller is expressed as:
in the method, in the process of the application,is an approximate optimal virtual controller of a single-link mechanical arm system,>is the optimal practical controller of the system approximation, eta is the designed barrier function, k c > 0 and k b > 0 is the upper and lower bound time-varying functions of the auxiliary function k, h is the designed conversion function, p, q, m and n are the error variables z derived from the barrier function eta 1 Conversion function h, upper and lower bound time-varying function k of auxiliary function k c And k b The former coefficients, and hasAnd beta 2 > 0 is a design parameter, z 2 Is an error variable, neural network->Approximation->Unknown dynamics in->Is an ideal estimate of the weight of the execution neural network,/->Then is in vector->z 1 Is an input Gaussian radial basis function; whereas neural networksFor approximation->Unknown dynamics in->An estimate representing the ideal weight of the executive neural network,then is in vector->z 2 Is an input gaussian radial basis function.
Preferably, in response to the process of performing adaptive optimal backstepping control on the single-link robot mechanical arm, the adaptive update rate of the execution-judgment neural network weight is designed based on a gradient descent method and a judgment method of the Lyapunov function stability theory, and the adaptive optimal backstepping control is performed on the single-link robot mechanical arm according to an optimal controller.
The application discloses a self-adaptive optimal backstepping control system for a single-connecting-rod mechanical arm, which comprises the following components:
the data acquisition module is used for acquiring the physical characteristics of the mechanical arm of the single-link robot;
the data processing module is used for converting the model into a state model by acquiring a dynamic model of the mechanical arm of the single-link robot based on the physical characteristics;
the control module is used for designing an optimal controller for the single-link robot mechanical arm by establishing an expected track model of a reference signal according to a state model and giving time-varying partial position constraint conditions of the joint angular displacement of the single-link robot and combining a backstepping technique and a reinforcement learning algorithm to carry out self-adaptive optimal backstepping control on the single-link robot mechanical arm.
Preferably, the control module is configured to obtain a desired trajectory model, where the desired trajectory model is expressed as:
y r ＝0.1(tanh(t-T s )+tanh(t-T r ))sin(t)+0.1cos(t)
wherein T represents time, T s Indicating the moment of onset of constrained joint angular displacement g, T r Indicating the end time at which the joint angular displacement g is constrained.
Preferably, the control module outputs g and the reference trajectory y according to a given single link robot arm system r Generating time-varying partial position constraints, wherein the constraints are:
(1) When t is E [ t ] 0 ,T s ) When the joint angular displacement is not restrained;
(2) When T is E [ T ] s ,T r ) When the joint angular displacement is constrained, and k is satisfied l ＜g＜k u ；
(3) When T is E [ T ] r Infinity), the angular displacement of the joint is not constrained;
wherein t is 0 Representing the starting instant, k, of the system operation u And k l Representing the constrained upper and lower time-varying functions of the joint angular displacement g, respectively.
Preferably, the control module constructs an optimal controller by designing an optimal virtual controller and an optimal actual controller according to a system dynamic equation of the dynamic model, wherein the optimal controller is expressed as:
in the method, in the process of the application,is an approximate optimal virtual controller of a single-link mechanical arm system,>is the optimal practical controller of the system approximation, eta is the designed barrier function, k c > 0 and k b > 0 is the upper and lower bound time-varying functions of the auxiliary function k, h is the designed conversion function, p, q, m and n are the error variables z derived from the barrier function eta 1 Conversion function h, upper and lower bound time-varying function k of auxiliary function k c And k b The former coefficients, and hasAnd beta 2 > 0 is a design parameter, z 2 Is an error variable, neural network->Approximation->Unknown dynamics in->Is an ideal estimate of the weight of the execution neural network,/->Then is in vector->z 1 Is an input Gaussian radial basis function; whereas neural networksFor approximation->Unknown dynamics in->An estimate representing the ideal weight of the executive neural network,then is in vector->z 2 Is an input gaussian radial basis function.
Preferably, the control module performs self-adaptive optimal backstepping control on the mechanical arm of the single-link robot by designing and executing self-adaptive update rate of the judgment neural network weight according to a gradient descent method and a stability judgment method of the Lyapunov function stability theory and according to the constructed optimal controller.
The application discloses the following technical effects:
1. the technical scheme of the application researches the tracking control problem of the single-link mechanical arm with time-varying part position limitation. Unlike common joint angular displacement, the present application requires that the single-link mechanical arm system can select whether the joint angular displacement needs to meet the position limiting constraint condition or not or even does not meet the position limiting constraint condition in the whole system operation process according to the actual engineering requirement at any time in the system operation process. Based on the requirements, the system which is required to meet the position limiting requirement in the whole course is converted into the system which is only required to meet the limiting requirement at a specific moment by designing a conversion function and a special barrier function, so that the constraint condition of time-varying position limitation of the joint angular displacement part of the single-link mechanical arm system is realized.
2. The technical scheme of the application provides a self-adaptive optimal backstepping control strategy. By designing the optimal performance index function containing the special obstacle function, the designed optimal controller can meet the limited position requirement of the time-varying part of the system joint angular displacement while realizing the optimal control target.
3. According to the technical scheme, the learning capability of the reinforcement learning algorithm is utilized, the adaptive update rate of the judging neural network and the executing neural network is designed by utilizing the gradient descent method, and the designed approximate optimal controller can approach to the actual optimal controller on the premise of stable system.
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In order to more clearly illustrate the embodiments of the present application or the technical solutions in the prior art, the drawings that are needed in the embodiments will be briefly described below, and it is obvious that the drawings in the following description are only some embodiments of the present application, and other drawings may be obtained according to these drawings without inventive effort for a person skilled in the art.
FIG. 1 is a flow chart of an implementation of a control method for a single-link mechanical arm according to an embodiment of the present application;
FIG. 2 is a schematic illustration of an actual single link robotic arm system;
FIG. 3 is a graph of joint angular position limitation versus reference trajectory tracking at some time after system operation;
FIG. 4 is a graph of joint angular position limitation and reference trajectory tracking for a period of time beginning system operation
FIG. 5 is a graph of system travel joint angular position unrestricted and reference trajectory tracking;
FIG. 6 is a graph of tracking error for a single link robotic arm system;
FIG. 7 is a graph of control inputs for a single link robotic arm system;
FIG. 8 is a graph of weight convergence for performing a neural network;
fig. 9 is a graph of the convergence of weights for evaluating neural networks.
Detailed Description
For the purpose of making the objects, technical solutions and advantages of the embodiments of the present application more apparent, the technical solutions of the embodiments of the present application will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present application, and it is apparent that the described embodiments are only some embodiments of the present application, not all embodiments. The components of the embodiments of the present application generally described and illustrated in the figures herein may be arranged and designed in a wide variety of different configurations. Thus, the following detailed description of the embodiments of the application, as presented in the figures, is not intended to limit the scope of the application, as claimed, but is merely representative of selected embodiments of the application. All other embodiments, which can be made by a person skilled in the art without making any inventive effort, are intended to be within the scope of the present application.
1-9, the application provides a self-adaptive optimal backstepping control method of a single-link mechanical arm with time-varying part limited positions, the whole flow is shown in FIG. 1, a single-link robot mechanical arm system is taken as an example for explaining the method in detail, and the detailed implementation process comprises the following steps:
step one: establishing a dynamics model of the mechanical arm of the single-link robot:
wherein, the liquid crystal display device comprises a liquid crystal display device,
j is moment of inertia, M is connecting rod mass, M 0 Is the load mass, L 0 Is the length of the connecting rod, R 0 Is the radius of the load, G is the gravity coefficient, B 0 Is the viscous friction coefficient at the joint, g (t) is the joint angular displacement, I (t) is the armature current of the motor, k τ Is a characteristic coefficient of electromechanical conversion of armature current into torque.
Let x 1 ＝g，u=i, then the system dynamics equation can be written as
Wherein x is 1 ,x 2 Representing a measurable system state.
The parameters defining the single link robot arm model in this embodiment are m=1, b=1 and n=10, respectively.
The expected trajectory model for a given reference signal in this embodiment is:
y r ＝0.1(tanh(t-T s )+tanh(t-T r ))sin(t)+0.1cos(t).
step two: and (3) giving time-varying partial position constraint conditions for the angular displacement of the joints of the single-link robot:
according to the output g and the reference track y of a given single-link robot arm system r The design output g satisfies the following constraint:
(1) When t is E [ t ] 0 ,T s ) When the joint angular displacement is not restrained;
(2) When T is E [ T ] s ,T r ) When the joint angular displacement is constrained, and k is satisfied l ＜g＜k u ；
(3) When T is E [ T ] r Infinity), the angular displacement of the joint is not constrained;
wherein t is 0 Indicating the moment of start of system operationTo be sure that t is not lost in generality 0 Equal to 0, T s Indicating the moment of onset of constrained joint angular displacement g, T r Indicating the end time of the constrained joint angular displacement g and having 0.ltoreq.t 0 ≤T s ＜T r ＜∞，k u And k l Representing the constrained upper and lower time-varying functions of the joint angular displacement g, respectively.
Designing a conversion function:
wherein v 1 And v 2 Is positive and has a finite constantAnd->Satisfy->And->n is the system order;
design error transfer function:
wherein z is 1 Is an error variable which is used to determine the error,is an optimal virtual controller->Is a function of the estimated value of (2);
designing an auxiliary function:
designing an obstacle function:
wherein k is c > 0 and k b > 0 represents the upper and lower bound time-varying functions of the auxiliary function k, respectively.
Step three: combining a back-stepping method technology and a reinforcement learning algorithm, designing an optimal controller:
respectively designing an optimal virtual controller and an optimal actual controller according to a system dynamic equation of a dynamic model of a given single-link mechanical arm system;
the first step:
the following optimal performance index function is established:
wherein Ω 1 Is a tight set containing the origin, ψ (Ω 1 ) Is an admission control set, alpha 1 Is a virtual controller which is used for controlling the operation of the computer,is the optimal virtual controller,>is a cost function.
The two sides of the formula are simultaneously derived to obtain the Hamiltonian Jacobian (HJB) equation as follows:
by solving forThe optimal virtual controller is obtained as follows:
will beThe method comprises the following steps of:
wherein, the liquid crystal display device comprises a liquid crystal display device,
is a design parameter.
Thus, the optimal virtual controller can be expressed as
Reinforcement learning algorithm based on executive-judgment neural network is adopted for respectively matching/dz 1 Approximation with the optimal virtual controller
Wherein, the liquid crystal display device comprises a liquid crystal display device,is->Estimated value of ∈10->Is->Is the estimated value of (2), neural network->Approximation is madeUnknown dynamics in->Is an ideal estimated value for judging the weight of the neural network, and the neural network is +.>Approximation->Unknown dynamics in->Is an ideal estimate of the weight of the execution neural network,/->Then is in vector->z 1 Is an input Gaussian radial basis function;
according to the above formula, the approximated HJB function is as follows:
the bellman residual is defined as follows:
and a second step of:
the optimum performance index function is established as follows:
where u is the actual controller, u * Is an optimal practical controller for the control system,is a cost function.
The corresponding Hamiltonian Jacobian (HJB) equation is as follows:
similar to the first step, the approximate optimal actual controller is obtained by adopting reinforcement learning algorithm
Wherein, the liquid crystal display device comprises a liquid crystal display device,is u * Is the estimated value of (2), neural network->For approximation->Unknown dynamics in->Estimated value representing ideal weight of the executive neural network,/->Then is in vector->z 2 Is an input gaussian radial basis function.
Step four: the self-adaptive updating rate of the execution-judgment neural network weight is designed based on a gradient descent method and a stability judgment method of the Lyapunov function stability theory:
wherein, gamma c1 And gamma c2 Is used for judging the design parameters of the neural network, gamma a1 And gamma a2 Is a design parameter for implementing neural networks.
To demonstrate the feasibility, effectiveness and correctness of this example, the present application performed the following simulation experiments:
in the simulation experiment, the self-adaptive optimal controller based on reinforcement learning is designed for the single-link mechanical arm system with the time-varying part position limitation, so that the joint angle position of the single-link mechanical arm system can track an ideal reference track, and meanwhile, the requirement of the time-varying part position limitation can be met.
In the design process of the optimal controller, judging the neural networkAnd->And execute neural network->Andall contain 6 neurons, and the central points of the neural network are uniformly distributed in [ -3,3]The width is:the initial weights of the neural network are: />When T is s ≤t＜T r When the error variable z 1 Is: k (k) b ＝k c =0.03+0.001sint, reference signal y r Is: y is l ＝-0.12,y u =0.12, so, system state g 1 Is: k (k) l ＝-k b +y l ＝-0.15-0.001sint,k u ＝k c +y u ＝0.15+0.001sint。
In the simulation process, three cases are discussed,
the first case is that the constraint of the system position limitation occurs between 10s and 30s in the running process of the system, and the initial state value of the system is given as follows: g 1 (0)＝0.08,g 2 (0) =0.1; the design parameters of the system are respectively as follows: beta i ＝80,γ ai ＝1.8,γ ci ＝1.1,υ i ＝0.1,i＝1,2；
The second case is that the constraint of the system position limitation occurs between 0s and 30s in the running process of the system, and the initial state value of the system is given as follows: g 1 (0)＝0.08,g 2 (0) =0.1; the design parameters of the system are respectively as follows: beta 1 ＝100,β 2 ＝60,γ ai ＝1.8,γ ci ＝1.1,υ i ＝0.1,i＝1,2；
The third condition is that the constraint of the system position limitation does not occur in the whole system operation process, and the initial state value of the system is given as follows: g 1 (0)＝0.08,g 2 (0)＝0.1；
The design parameters of the system are respectively as follows: beta 1 ＝80,β 2 ＝60,γ ai ＝1.8,γ ci ＝1.1,υ i ＝0.1,i＝1,2。
Simulating a mathematical model established in the control method of the embodiment by using MATLAB software, wherein FIG. 2 is a schematic diagram of an actual single-link mechanical arm system, simulation results are shown in FIG. 3-FIG. 9, FIG. 3, FIG. 4 and FIG. 5 respectively describe a first case, a second case and a third case, wherein the joint angle positions of the single-link mechanical arm system are limited, and an output track and a reference track map, so that the joint angle positions of the single-link mechanical arm system are respectively restrained in a designated area between 10s and 30s and between 0s and 30s of the system operation, and meanwhile, accurate tracking of reference signals is realized; FIG. 6 shows a graph of tracking error for a single link robotic arm system, from which it was found that the tracking error converged into the domain where the origin is relatively small, demonstrating the effectiveness of the proposed strategy of the present application; FIG. 7 illustrates a control input graph for a single link robotic arm system; fig. 8 and 9 show the weight convergence graphs of the execution neural network and the judgment neural network, respectively, from which it can be seen that each weight in the execution neural network and the judgment neural network eventually tends to be stable and converged, so that the designed reinforcement learning neural network control scheme can effectively achieve accurate approximation of the designed optimal controller, and has good convergence effect; thus, simulations demonstrate the effectiveness of the proposed control scheme.
In the embodiment, a back-step recursion and reinforcement learning technology is taken as a design framework, and the self-adaptive optimal back-step control problem is researched aiming at a nonlinear single-link mechanical arm system with a time-varying part with limited position, so that the system control requirement is met on the basis of reducing energy consumption. In addition, the application realizes the optimal control by designing the transfer function and the barrier function and combining the optimal control and the backstepping method technology, and simultaneously meets the requirement of limited position of the time-varying part of the system, so that the system has more generality. According to the embodiment, the approximation capability of the reinforcement learning neural network to the nonlinear function is utilized, the reinforcement learning algorithm is utilized to approximate the actual optimal controller, the problem that the HJB equation is difficult to directly solve is solved, the optimal controller meeting the control requirement is designed, the weight of the neural network is adjusted through the self-adaption rate, and the designed approximate optimal controller can approximate the actual optimal controller. Finally, the self-adaptive optimal backstepping control strategy proposed by simulation verification ensures the quality of all signals. The popularization and application of the application in the limited tracking control of the position of the single-link mechanical arm are one of important research directions in the future.
The present application is described with reference to flowchart illustrations and/or block diagrams of methods, apparatus (systems) and computer program products according to embodiments of the application. It will be understood that each flow and/or block of the flowchart illustrations and/or block diagrams, and combinations of flows and/or blocks in the flowchart illustrations and/or block diagrams, can be implemented by computer program instructions. These computer program instructions may be provided to a processor of a general purpose computer, special purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create means for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.
In the description of the present application, it should be understood that the terms "first," "second," and the like are used for descriptive purposes only and are not to be construed as indicating or implying a relative importance or number of technical features indicated. Thus, a feature defining "a first" or "a second" may explicitly or implicitly include one or more such feature. In the description of the present application, the meaning of "a plurality" is two or more, unless explicitly defined otherwise.
It will be apparent to those skilled in the art that various modifications and variations can be made to the present application without departing from the spirit or scope of the application. Thus, it is intended that the present application also include such modifications and alterations insofar as they come within the scope of the appended claims or the equivalents thereof.
Claims (10)
1. An adaptive optimal backstepping control method for a single-link mechanical arm, comprising the steps of:
obtaining a dynamic model of a single-link robot mechanical arm, and converting the model into a state model by obtaining physical characteristics of the single-link robot mechanical arm;
based on the state model, an optimal controller for the single-link robot mechanical arm is designed by establishing an expected track model of a reference signal and giving a time-varying partial position constraint condition of the joint angular displacement of the single-link robot and combining a backstepping technique and a reinforcement learning algorithm, and the single-link robot mechanical arm is subjected to self-adaptive optimal backstepping control.
2. The adaptive optimal backstepping control method for a single link mechanical arm according to claim 1, wherein:
in response to the establishment of the desired trajectory model, the desired trajectory model is represented as:
y r ＝0.1(tanh(t-T s )+tanh(t-T r ))sin(t)+0.1cos(t)
wherein T represents time, T s Indicating the moment of onset of constrained joint angular displacement g, T r Indicating the end time at which the joint angular displacement g is constrained.
3. The adaptive optimal backstepping control method for a single link mechanical arm according to claim 2, wherein:
in response to the process of obtaining the time-varying partial position constraint condition, according to the given single-link robotic arm system output g and the reference trajectory y r Generating the time-varying partial position constraint condition, wherein the constraint condition is:
(1) When t is E [ t ] 0 ,T s ) When the joint angular displacement is not restrained;
(2) When T is E [ T ] s ,T r ) When the joint angular displacement is constrained, and k is satisfied l ＜g＜k u ；
(3) When T is E [ T ] r Infinity), the angular displacement of the joint is not constrained;
wherein t is 0 Representing the starting instant, k, of the system operation u And k l Representing the constrained upper and lower time-varying functions of the joint angular displacement g, respectively.
4. A self-adaptive optimal backstepping control method for a single link mechanical arm according to claim 3, wherein:
in response to a design process of the optimal controller, respectively designing an optimal virtual controller and an optimal actual controller based on a system dynamic equation of the dynamic model, and constructing the optimal controller, wherein the optimal controller is expressed as:
in the method, in the process of the application,is an approximate optimal virtual controller of a single-link mechanical arm system,>is the optimal practical controller of the system approximation, eta is the designed barrier function, k c > 0 and k b > 0 is the upper and lower bound time-varying functions of the auxiliary function k, h is the designed conversion function, p, q, m and n are the error variables z derived from the barrier function eta 1 Conversion function h, upper and lower bound time-varying function k of auxiliary function k c And k b The former coefficients, and hasβ 1 > 0 and beta 2 > 0 is a design parameter, z 2 Is an error variable, neural network->Approximation->Unknown dynamics in->Is an ideal estimate of the weight of the execution neural network,/->Then is in vector->z 1 Is an input Gaussian radial basis function; whereas neural networksFor approximation->Unknown dynamics in->An estimate representing the ideal weight of the executive neural network,then is in vector->z 2 Is an input gaussian radial basis function.
5. The adaptive optimal backstepping control method for a single link mechanical arm according to claim 4, wherein:
and responding to the process of performing self-adaptive optimal backstepping control on the mechanical arm of the single-link robot, designing and executing the self-adaptive updating rate of the judgment neural network weight based on a gradient descent method and a judgment method of the Lyapunov function stability theory, and performing self-adaptive optimal backstepping control on the mechanical arm of the single-link robot according to the optimal controller.
6. An adaptive optimal backstepping control system for a single link robotic arm, comprising:
the data acquisition module is used for acquiring the physical characteristics of the mechanical arm of the single-link robot;
the data processing module is used for converting the model into a state model by acquiring a dynamic model of the mechanical arm of the single-link robot based on the physical characteristics;
the control module is used for designing an optimal controller for the single-link robot mechanical arm by establishing an expected track model of a reference signal according to the state model and giving time-varying partial position constraint conditions of the joint angular displacement of the single-link robot, combining a backstepping technique and a reinforcement learning algorithm, and performing self-adaptive optimal backstepping control on the single-link robot mechanical arm.
7. The adaptive optimal backstepping control system for a single link robotic arm of claim 6, wherein:
the control module is configured to obtain the desired trajectory model, where the desired trajectory model is expressed as:
y r ＝0.1(tanh(t-T s )+tanh(t-T r ))sin(t)+0.1cos(t)
wherein T represents time, T s Indicating the moment of onset of constrained joint angular displacement g, T r Indicating the end time at which the joint angular displacement g is constrained.
8. An adaptive optimal backstepping control system for a single link robotic arm as defined in claim 7, wherein:
the control module outputs g and a reference track y according to a given single-link robot mechanical arm system r Generating the time-varying partial position constraint condition, wherein the constraint condition is:
(1) When t is E [ t ] 0 ,T s ) When the joint angular displacement is not restrained;
(2) When T is E [ T ] s ,T r ) When the joint angular displacement is constrained, and k is satisfied l ＜g＜k u ；
(3) When T is E [ T ] r Infinity), the angular displacement of the joint is not constrained;
wherein t is 0 Representing the starting instant, k, of the system operation u And k l Representing the constrained upper and lower time-varying functions of the joint angular displacement g, respectively.
9. An adaptive optimal backstepping control system for a single link robotic arm as defined in claim 8, wherein:
the control module constructs an optimal controller by respectively designing the optimal virtual controller and the optimal actual controller according to a system dynamic equation of the dynamic model, wherein the optimal controller is expressed as:
in the method, in the process of the application,is an approximate optimal virtual controller of a single-link mechanical arm system,>is the optimal practical controller of the system approximation, eta is the designed barrier function, k c > 0 and k b > 0 is the upper and lower bound time-varying functions of the auxiliary function k, h is the designed conversion function, p, q, m and n are the error variables z derived from the barrier function eta 1 Conversion function h, upper and lower bound time-varying function k of auxiliary function k c And k b The former coefficients, and hasβ 1 > 0 and beta 2 > 0 is a design parameter, z 2 Is an error variable, neural network->Approximation->Unknown dynamics in->Is an ideal estimate of the weight of the execution neural network,/->Then is in vector->z 1 Is an input Gaussian radial basis function; whereas neural networksFor approximation->Unknown dynamics in->An estimate representing the ideal weight of the executive neural network,then is in vector->z 2 Is an input gaussian radial basis function.
10. An adaptive optimal backstepping control system for a single link robotic arm as defined in claim 9, wherein:
the control module performs self-adaptive optimal backstepping control on the mechanical arm of the single-link robot according to the gradient descent method and the stability judging method of the Lyapunov function stability theory by designing and executing-judging the self-adaptive update rate of the neural network weight and the constructed optimal controller.
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