CN113211446B - Mechanical arm decentralized tracking control method for event triggering-nerve dynamic programming - Google Patents
Mechanical arm decentralized tracking control method for event triggering-nerve dynamic programming Download PDFInfo
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Abstract
A mechanical arm decentralized tracking control method for event triggering-nerve dynamic programming comprises the following steps: the method comprises the steps of constructing a modularized mechanical arm subsystem dynamics model based on a joint moment feedback technology, introducing an event trigger mechanism by designing an improved cost function of comprehensive tracking performance, controller output and approximate model items, updating a system control law if and only if a trigger condition is met, approximately solving a Hamiltonian equation based on the event trigger mechanism by utilizing an evaluation neural network, and finally obtaining a modularized mechanical arm joint module subsystem nerve-optimal tracking control strategy based on the event trigger mechanism, so that the mechanical arm system is ensured to safely and stably run when contacting with the external environment.
Description
Technical Field
The invention relates to a modularized mechanical arm scattered tracking control method for event triggering-nerve dynamic programming, belonging to the field of robot control systems and control algorithms.
Background
The modularized mechanical arm is a mechanical arm with standard modules and interfaces, and can be used for recombining and configuring the self configuration according to different task requirements. According to the modular concept, the joint module of the modular mechanical arm comprises units of communication, driving, control, sensing and the like, so that the mechanical arm can change the configuration according to task requirements under different external environments and constraints, and the reconstructed mechanical arm can have better adaptability to new working environments. The modularized mechanical arm is one of the most promising robots because of the characteristics of easy assembly and carrying, strong adaptability, low cost and the like, and is hopefully applied to many scenes in which human beings cannot directly participate, such as intelligent manufacturing, deep space exploration, high risk rescue and other task environments. In order to complete the task work under such complex extreme environments, besides ensuring the control performance thereof, the calculation amount, the communication width and the energy consumption are also key problems which need to be concerned and solved.
The event triggering mechanism is derived from a network communication control system, and refers to whether the control task is executed or not, which is determined by a given triggering condition, and the execution of the control task is non-periodic and not updated according to time. If the trigger condition violates at some point, this means that the event triggers and immediately executes the control task. Compared with the periodic updating of the time triggering mechanism according to the time variation, the event triggering mechanism can effectively reduce the execution times of the control task while guaranteeing the stability of the closed-loop system, and obviously saves communication resources. For the mechanical arm system, the execution times of the controller are the update times of the actuator, that is to say, the event triggering control can effectively reduce the update times of the actuator, thereby achieving the control aim of protecting the actuator and saving energy. At present, the research on the event trigger control method of the mechanical arm is less, and mainly because the mechanical arm is a hardware system with strong real-time performance, the unknown behaviors such as locking or trembling of the mechanical arm can be directly caused by unsuitable event trigger conditions. In addition, for a modular robotic arm system, because of its configuration changing with task demands, its dynamics model is difficult to build with conventional methods, making it more difficult for event-triggered control algorithms for individual joint subsystems to advance. How to design corresponding event triggering conditions for a joint module subsystem by using measurable or known joint model information, and greatly reduce the output energy consumption and the calculated amount of the system on the premise of ensuring the safe and stable operation of the system is a problem to be solved.
In order to ensure that a modularized mechanical arm under the condition of limited energy sources has good stability and accuracy, a modularized mechanical arm subsystem dynamics model based on a joint moment feedback technology is established, an event trigger mechanism is introduced by designing an improved cost function of comprehensive tracking performance, controller output and approximate model items, if and only when trigger conditions are met, a system control law is updated, an evaluation neural network is utilized to approximately solve a Hamiltonian equation based on the event trigger mechanism, and finally, a modularized mechanical arm joint module subsystem nerve-optimal tracking control strategy based on the event trigger mechanism is obtained, so that calculation redundancy and storage space are reduced while the fact that an actual position can track an expected track is ensured, and communication resources and energy consumption are optimized.
Disclosure of Invention
In order to solve the problems in the traditional modularized mechanical arm tracking control system and the method thereof, the invention provides a mechanical arm scattered tracking control method for event triggering-nerve dynamic programming.
The technical scheme for solving the technical problems is as follows:
a mechanical arm decentralized tracking control method for event triggering-nerve dynamic programming is characterized by constructing a modularized mechanical arm subsystem dynamics model based on a joint moment feedback technology, introducing an event triggering mechanism by designing an improved cost function of comprehensive tracking performance, controller output and approximate model items, updating a system control law if and only if a triggering condition is met, and utilizing an evaluation neural network to approximately solve a Hamiltonian equation based on the event triggering mechanism to finally obtain a modularized mechanical arm joint module subsystem nerve-optimal tracking control strategy based on the event triggering mechanism.
An event triggering-nerve dynamic programming mechanical arm decentralized tracking control method comprises the following steps:
1. establishment of dynamic model
Consider an n-degree-of-freedom modular robotic arm system whose dynamics model using joint moment feedback techniques is expressed as:
wherein,joint position, velocity and acceleration for a modular robotic arm system; q i Is the ith joint position; />Is the angular velocity of the joint; />Is joint angular acceleration; i mi The motor rotational inertia; gamma ray i Is the reduction ratio of the speed reducer;kinetic coupling cross-links between the joints; />Is a friction torque term; τ fi Moment information measured by a joint moment sensor; τ i And outputting torque for the motor.
From the above-described deductive simplified arrangement, the dynamics model of the ith joint of an n-degree-of-freedom modular manipulator system is represented as a state space form of the following nonlinear system:
wherein,representing the articulation angular position and angular velocity states for the state vector of the modular robotic arm, +.>In the form of differentiation of system state with respect to time, y i D, outputting the modularized mechanical arm system i ＝(I mi γ i ) -1 ∈R + For conversion into rotational inertia terms of joints, u ei ＝τ i For the control moment of the ith joint and the input moment of the actuator, Γ fi Is alreadyKnowing model term and Θ i Modeling errors and approximation errors for the model uncertainty term include friction torque models, and kinetic coupling cross-terms between joints, expressed as:
wherein f bi Is a constant to be determined; f (f) si Is a static friction related parameter; f (f) τi Modeling errors for position dependent friction and friction; f (f) ci Is a coulomb friction related parameter; f (f) pi (x 1i ,x 2i ) Is a non-parameterized friction term;as a sign function related to joint velocity; />For an approximation error vector consisting of the estimated errors of the friction terms,/->Respectively corresponding friction coefficient f bi ，f ci ，f si ，f τi Is used for the estimation of the (c),a vector consisting of functions related to sub-joint position and sub-joint velocity; />For kinetic coupling cross-links between joints, < ->The position, the speed and the acceleration direction of the modularized mechanical arm respectivelyAmount of the components.
2. Adaptive dynamic programming algorithm for time triggered mechanism
In order to realize the optimal control work target of the modularized mechanical arm system under the limited energy, aiming at a joint subsystem dynamics model based on a joint moment sensor feedback technology, firstly, establishing a performance index function of a comprehensive control target under a time trigger mechanism:
wherein θ i (x i ) Including joint position tracking error and velocity tracking error θ for a hybrid tracking error function i (x i )＝x 2i -x 2di +a ei (x 1i -x 1di )，a ei For a given coefficient, x 1di And x 2di Representing a desired position tracking target and a velocity tracking target, respectively, an initial value θ i0 (x i (0))＝θ i (0) Matrix Q ei And R is ei Is a positive definite matrix of the matrix and the matrix,as an effect function, give an initial value N ei (0, 0) =0, where u ei (θ i )＝u ei (θ i (x i (t))) by combining u ei (x i ) Conversion to u ei (θ i ) And a distributed optimal control strategy with higher control precision is realized. Psi i (Ω e ) Is a set of possible control strategies, Ω e For a given set. G i As the modular mechanical arm is a practical system, the uncertainty factors such as friction terms and cross-linking terms of the model existing in the modular mechanical arm only slightly change along with the temperature or the lubricity, so that the inequality is satisfied>
According to the proposed performance index function, the Hamiltonian-Jacobian-Belman equation of the joint subsystem under the time trigger mechanism is established as follows:
wherein,representing xi i (θ i ) For theta i Partial derivative form> As a function of the desired joint acceleration and the actual joint velocity. The optimal performance index function based on the time trigger mechanism is as follows:
by combining with the optimization thought, the optimal performance index functionSatisfying the Hamiltonian equation:
if the optimum performance index functionIs continuously differentiable and satisfies the above equation, the corresponding Hamiltonian presence solution of a nonlinear continuous system is used as an optimal control strategy +.>Can be expressed as:
3. optimal control of modularized mechanical arm under event triggering mechanism
Based on the event trigger mechanism principle, the tracking control law input of the modular robotic system is calculated and updated if and only if the trigger condition is violated. Assume thatIs a group of monotone increasing sequences composed of trigger moments, which satisfies 0 < t l ＜t l+1 And there is +.f. for l ε {0,1,2, … }>Defining the sampling moment state as follows:
wherein,for t E [ t ] l ,t l+1 ) And sampling the data information of the state in time. To obtain a suitable event triggering condition, an interval function E is defined which represents the relation between the sampled state and the actual state information li (t) is:
according to the event trigger mechanism, the control law is updated following the trigger condition, that is, the state at this time is expressed as:
under an event triggering mechanism, in order to ensure the safe operation of the modularized mechanical arm, the tracking control law changes a discrete signal into a continuous signal through a zero-order retainer, and the expression is as follows:
at t E [ t ] l ,t l+1 ) Within the time of day. According to the optimal control strategy under the time trigger mechanism, defining the optimal control strategy of the event trigger mechanism as follows:
however, the process is not limited to the above-described process,discrete values sampled non-periodically, the control signal is changed from discrete to continuous by introducing a zero-order keeper.
4. Construction of judgment neural network
To solve Hamiltonian based on event trigger mechanism, the strong learning ability of neural network is utilized to approximate performance index function (XI) i (θ i )：
Wherein,is an ideal weight, N i For the number of hidden layer neurons, +.>Is an activation function, ε ei (θ i ) To evaluate the neural network approximation residuals. Performance index function based on event trigger mechanism +.>The partial derivative of (2) is expressed as:
wherein,and->The activation function and the partial derivative of the estimated neural network approximation residual are respectively. Let->And->δ eid And epsilon eid Is a positive constant. From the above description, the inequality is deduced +.>Is true, where P ei For a given control parameter.
However, due to the ideal evaluation neural network weight W ei We cannot directly learn, so approximate the evaluation neural network as:
wherein,is an approximation of the weights of the neural network. Thus, an approximate event-triggered distributed tracking control strategy is obtained:
the approximate Hamiltonian is derived as:
wherein ε EHi Is an estimated residual of the approximated event-triggered hamiltonian. Next, an evaluation neural network weight estimation error is defined:in order to evaluate the estimated weights of the neural network +.>Tracked upper expected weight W ei Minimizing the objective function by using gradient descent algorithm>Adjusting the weight vector of the neural network>Its update strategy is designed:
wherein alpha is ei To evaluate the update law of network weight, designIs a function related to the modular robotic system state and neural network activation functions.
Finally, an approximate event-triggered distributed tracking control strategy is obtained:
according to the Lyapunov stability theorem and an event triggering mechanism, an event triggering condition which can stabilize the system is designed for the modularized mechanical arm joint module subsystem with n degrees of freedom, namely:
wherein lambda is min () And lambda (lambda) max () Representing the minimum eigenvalue and the maximum eigenvalue of the matrix in brackets, respectively. E (E) LEi Event triggering conditions for dispersing optimal tracking control are triggered for the proposed modular mechanical arm events.
The beneficial effects of the invention are as follows:
in the aspect of mechanical arm track tracking control, the invention solves the problem of force/position tracking task targets under the condition of uncertainty factors facing to the constraint working space, and adopts the self-adaptive estimation algorithm to approximate the unknown constraint parameter vector on line so as to ensure that the mechanical arm system runs safely and stably when contacting with the external environment.
In the aspect of control precision, the invention converts a force/position control task into a corresponding optimal control problem to solve, and introduces an adaptive dynamic programming method, so that the contact force and position tracking performance of the end effector are more continuous and smooth, and the output consumption of the actuator is reduced while the progressive stability of the system is maintained.
Therefore, the invention solves the problems of low response speed and low tracking precision of force/position control facing to the influence of unknown environmental uncertainty factors in the prior art, provides stability and precision for the modularized mechanical arm, and can meet the requirements of contact tasks with various complex environments.
Drawings
Fig. 1 is a schematic diagram of a distributed tracking control of a mechanical arm for event triggering-neural dynamic programming according to the present invention.
Detailed Description
The invention is described in further detail below with reference to the accompanying drawings.
As shown in fig. 1, the invention relates to a mechanical arm decentralized tracking control method for event triggering-nerve dynamic programming, which comprises the following specific implementation method and process:
1. establishment of dynamic model
Consider an n-degree-of-freedom modular robotic arm system whose dynamics model using joint moment feedback techniques is expressed as:
wherein q i Is the ith joint position;is the angular velocity of the joint; />Is joint angular acceleration; i mi The motor rotational inertia; gamma ray i Is the reduction ratio of the speed reducer; />Kinetic coupling cross-links between the joints; τ fi Moment information measured by a joint moment sensor; τ i The torque is output for the motor; />As a friction torque term, a class of functions related to joint position and joint velocity will be defined:
wherein f bi Is a constant to be determined; f (f) si Is a static friction related parameter; f (f) τi Modeling errors for position dependent friction and friction; f (f) ci Is a coulomb friction related parameter;is a non-parameterized friction term. Furthermore, the symbolNumber function->Is defined as:
taking into account friction momentThe friction between flexible gears in the harmonic reducer and friction in the executing motor of each joint module are mainly included. As known from the prior researches of a large number of scholars, the established model (2) of the friction moment term is effective and obtained through experimental verification, and the nominal coefficient f can be obtained bi ，f ci ，f si ，f τi Seen as being quite close to the actual value. Thus, according to the linearization criterion, the friction term to which the joint is subjected is estimated by the following equation:
wherein,for an approximation error vector consisting of the estimated errors of the friction terms,/->Respectively corresponding friction coefficient f bi ，f ci ，f si ，f τi Is used for the estimation of the (c),a vector consisting of functions related to joint position and joint velocity.
From the above-described deductive simplified arrangement, the dynamics model of the ith joint of an n-degree-of-freedom modular manipulator system is represented as a state space form of the following nonlinear system: :
wherein,representing the articulation angular position and angular velocity states for the state vector of the modular robotic arm, +.>In the form of differentiation of system state with respect to time, y i D, outputting the modularized mechanical arm system i ＝(I mi γ i ) -1 ∈R + For conversion into rotational inertia terms of joints, u ei ＝τ i For the control moment of the ith joint and the input moment of the actuator, Γ fi For known model terms and theta i Modeling errors and approximation errors for the model uncertainty term include friction torque models, and kinetic coupling cross-terms between joints, expressed as:
2. adaptive dynamic programming algorithm for time triggered mechanism
In order to realize the optimal control work target of the modularized mechanical arm system under the limited energy, aiming at a joint subsystem dynamics model based on a joint moment sensor feedback technology, firstly, establishing a performance index function of a comprehensive control target under a time trigger mechanism:
wherein θ i (x i (t)) is a hybrid tracking error function comprising joint position tracking error and velocity tracking error θ i (x i )＝x 2i -x 2di +a ei (x 1i -x 1di )，a ei For a given coefficient, x 1di And x 2di Representing a desired position tracking target and a velocity tracking target, respectively, an initial value θ i0 (x i (0))＝θ i (0) Matrix Q ei And R is ei Is a positive definite matrix of the matrix and the matrix,as an effect function, give an initial value N ei (0, 0) =0, where u ei (θ i )＝u ei (θ i (x i (t))) by combining u ei (x i ) Conversion to u ei (θ i ) And a distributed optimal control strategy with higher control precision is realized. Psi i (Ω e ) Is a set of possible control strategies, Ω e For a given set. Γ -shaped structure fi For known model terms, G i As the modular mechanical arm is a practical system, the uncertainty factors such as friction terms and crosslinking terms of the model existing in the modular mechanical arm only slightly change along with the temperature or the lubrication degree, thereby satisfying the inequality
For a modular robotic arm joint module subsystem, forIf there is a set of admission control strategies mu ei (θ i )∈Ψ i (Ω e ) A set of possible control strategies and fulfils μ ei (0) =0, if μ ei (θ i ) At Ω e Above is continuous, mu ei (θ i )＝u ei (θ i ) Can ensure that the modularized mechanical arm system is tightly assembled with omega e Up-convergence and ensure performance index function i (θ i ) Is bounded and in an initial state xi i (0)＝0。
According to the proposed performance index function, the Hamiltonian-Jacobian-Belman equation of the joint subsystem under the time trigger mechanism is established as follows:
wherein,representing xi i (θ i ) For theta i Partial derivative form> As a function of the desired joint acceleration and the actual joint velocity. The optimal performance index function based on the time trigger mechanism is as follows:
by combining with the optimization thought, the optimal performance index functionSatisfying the Hamiltonian equation:
if the optimum performance index functionIs continuously differentiable and satisfies the above equation, the corresponding Hamiltonian presence solution of a nonlinear continuous system is used as an optimal control strategy +.>Can be expressed as:
bringing the optimal control strategy into the Hamiltonian, deriving:
through the thought of optimal control, the self-adaptive dynamic programming algorithm can be utilized to solve the approximate optimal control strategy based on the time trigger mechanism. However, the time triggered mechanism is not only computationally intensive, traffic intensive, but also wastes limited energy resources. In order to overcome the above-mentioned shortcomings, an optimal control method based on an event trigger mechanism is proposed, wherein the sampling time of the control method is determined by a certain trigger condition, instead of a fixed sampling interval. The design of the control signal update can obviously save calculation and communication resources. Next, we will introduce a modular robotic arm optimal control mechanism based on event triggering.
3. Modular mechanical arm subsystem neural-optimal control under event triggering mechanism
Based on the event trigger mechanism principle, the tracking control law input of the modular robotic system is calculated and updated if and only if the trigger condition is violated. Assume thatIs a group of monotone increasing sequences composed of trigger moments, which satisfies 0 < t l ＜t l+1 And there is +.f. for l ε {0,1,2, … }>Defining the sampling moment state as follows:
wherein,for t E [ t ] l ,t l+1 ) And sampling the data information of the state in time. To obtain a suitable event triggering condition, an interval function E is defined which represents the relation between the sampled state and the actual state information li (t) is:
according to the event trigger mechanism, the control law is updated following the trigger condition, that is, the state at this time is expressed as:
under an event triggering mechanism, in order to ensure the safe operation of the modularized mechanical arm, the tracking control law changes a discrete signal into a continuous signal through a zero-order retainer, and the expression is as follows:
at t E [ t ] l ,t l+1 ) Within the time of day. According to the optimal control strategy under the time trigger mechanism, defining the optimal control strategy of the event trigger mechanism as follows:
however, the process is not limited to the above-described process,discrete values of the non-periodic samples, by introducing a zero-order holder,so that the control signal is changed from a discrete signal to a continuous signal. Combining the Hamiltonian equation based on the time trigger mechanism and the optimal control law based on the event trigger mechanism to obtain the Hamiltonian equation based on the event trigger mechanism, wherein the Hamiltonian equation based on the event trigger mechanism is as follows:
for all statesThere is a designed tracking control law that is Lipschitz continuous and there is a given positive constant m li The following relationship is satisfied:
tracking control strategy based on event trigger mechanism, tracking error equation of sub-joint module system according to trigger time interval t l Will be theta i (x i ) Is replaced byThe tracking control strategy is updated according to the event trigger condition
Construction of judgment neural network
To solve Hamiltonian based on event trigger mechanism, the strong learning ability of neural network is utilized to approximate performance index function (XI) i (θ i )：
Wherein,is an ideal weight, N i For the number of hidden layer neurons, +.>Is an activation function, ε ei (θ i ) To evaluate the neural network approximation residuals. Performance index function based on event trigger mechanism +.>The partial derivative of (2) is expressed as:
wherein,and->The activation function and the partial derivative of the estimated neural network approximation residual are respectively. Let->And->δ eid And epsilon eid Is a positive constant. From the above description, the inequality is deduced +.>Is true, where P ei For a given control parameter. Thus, the optimal control law is approximated as:
the Hamiltonian obtained based on the event trigger mechanism is expressed as:
however, due to the ideal evaluation neural network weight W ei We cannot directly learn, so approximate the evaluation neural network as:
wherein,is an approximation of the weights of the neural network. Thus, an approximate event-triggered distributed tracking control strategy is obtained:
the approximate Hamiltonian is derived as:
wherein ε EHi Is an estimated residual of the approximated event-triggered hamiltonian. Next, an evaluation neural network weight estimation error is defined:in order to evaluate the estimated weights of the neural network +.>Tracked upper expected weight W ei Minimizing the objective function by using gradient descent algorithm>Adjusting the weight vector of the neural network>Its update strategy is designed:
wherein alpha is ei To evaluate the update law of network weight, design
Finally, an approximate event-triggered distributed tracking control strategy is obtained:
according to the Lyapunov stability theorem and an event triggering mechanism, an event triggering condition which can stabilize the system is designed for the modularized mechanical arm joint module subsystem with n degrees of freedom, namely:
wherein lambda is min () And lambda (lambda) max () Representing the minimum eigenvalue and the maximum eigenvalue of the matrix in brackets, respectively. E (E) LEi Event triggering conditions for dispersing optimal tracking control are triggered for the proposed modular mechanical arm events.
The distributed tracking control method for the mechanical arm for event triggering-nerve dynamic programming can solve the problems of low response speed and low tracking precision of force/position control facing the influence of unknown environmental uncertainty factors in the prior art, provides stability and accuracy for the operation of a modularized mechanical arm, and can meet the requirements of contact tasks with various complex environments. In the present specification, each embodiment is described in a progressive manner, and each embodiment is mainly described in a different point from other embodiments, and identical and similar parts between the embodiments are all enough to refer to each other. Variations in the detailed description and the application scope will occur to those skilled in the art upon consideration of the teachings of the present invention. In view of the foregoing, this description should not be construed as limiting the invention.
Claims (1)
1. A mechanical arm decentralized tracking control method for event triggering-nerve dynamic programming comprises the following steps: constructing a modularized mechanical arm subsystem dynamics model based on a joint moment feedback technology, designing uncertainty of an approximate estimation model of an extended state observer, introducing an event trigger mechanism by designing an improved cost function of comprehensive tracking performance, controller output and approximate model items, updating a system control law if and only if a trigger condition is met, and approximately solving a Hamiltonian equation based on the event trigger mechanism by utilizing an evaluation neural network to finally obtain a modularized mechanical arm joint module subsystem nerve-optimal tracking strategy based on the event trigger mechanism;
the method comprises the following steps:
step one, consider an n-degree-of-freedom modular robotic arm system, and represent its kinetic model by using joint moment feedback technology as:
wherein, q is,joint position, velocity and acceleration for a modular robotic arm system; q i Is the ith joint position; />Is the angular velocity of the joint; />Is joint angular acceleration; i mi The motor rotational inertia; γi is the reduction ratio of the speed reducer; />Is a kinetic coupling cross-link between joints; τ fi Moment information measured by a joint moment sensor; τ i The torque is output for the motor; />As a friction torque term, it is defined as a class of functions related to joint position and joint velocity:
wherein f bi Is a constant to be determined; f (f) si Is a static friction related parameter; f (f) τi Modeling errors for position dependent friction and friction; f (f) ci Is a coulomb friction related parameter;is a non-parameterized friction term; furthermore, the sign function->Is defined as:
taking into account the total friction momentThe friction between flexible gears in the harmonic reducer and friction in the executing motor of each joint module are mainly included; the nominal coefficient f is calculated bi ，f ci ，f si ，f τi Seen as very close to the actual value; thus, according to the linearization criterion, the friction term to which the joint is subjected is estimated by the following equation:
wherein,for an approximation error vector consisting of the estimated errors of the friction terms,/->Respectively corresponding friction coefficient f bi ，f ci ，f si ，f τi Is used for the estimation of the (c),a vector consisting of functions related to joint position and joint velocity;
from the deductive simplified organization, the dynamics model of the ith joint of the n-degree-of-freedom modularized mechanical arm system is expressed as the state space form of the following nonlinear system:
wherein,representing the articulation angular position and angular velocity states for the state vector of the modular robotic arm, +.>In the form of differentiation of system state with respect to time, y i For the output of the modular robotic arm system, di= (I mi γ i ) -1 ∈R + For conversion into rotational inertia terms of joints, u ei ＝τ i For the control moment of the ith joint and the input moment of the actuator, Γ fi For known model terms and theta i Modeling errors and approximation errors for the model uncertainty term include friction torque models, and kinetic coupling cross-terms between joints, expressed as:
secondly, aiming at a joint subsystem dynamics model based on a joint moment sensor feedback technology, a performance index function of a comprehensive control target under a time trigger mechanism is firstly established for realizing a work target of optimal control of the modularized mechanical arm system under the condition of limited energy sources:
wherein θ i (x i (t)) is a hybrid tracking error function comprising joint position tracking error and velocity tracking error θ i (x i )＝x 2i -x 2di +a ei (x 1i -x 1di )，a ei For a given coefficient, x 1di And x 2di Representing a desired position tracking target and a velocity tracking target, respectively, an initial value θ i0 (x i (0))＝θ i (0) Matrix Q ei And R is ei Is a positive definite matrix of the matrix and the matrix,as an effect function, give an initial value N ei (0, 0) =0, where u ei (θ i )＝u ei (θi(x i (t))) by combining u ei (x i ) Conversion to u ei (θ i ) Realizing a distributed optimal control strategy with higher control precision; ψi (Ω) e ) Is a set of possible control strategies, Ω being a given set; Γ -shaped structure fi For known model terms, G i As the modular mechanical arm is a practical system, the uncertainty factors such as friction terms and cross-linking terms of the model existing inside the modular mechanical arm only slightly change with temperature or lubricity,thus satisfying inequality->For the modular manipulator joint module subsystem, for +.>If there is a set of admission control strategies mu ei (θ i )∈Ψi(Ω e ) A set of possible control strategies and fulfils μ ei (0) =0, if μ ei (θ i ) Is continuous at Ω e, μ ei (θ i )＝u ei (θ i ) Can ensure that the modularized mechanical arm system is tightly assembled with omega e Up-convergence and guarantee of the performance index function xi i (θ i ) Is bounded and in an initial state xi i (0)＝0；
According to the proposed performance index function, the Hamiltonian-Jacobian-Belman equation of the joint subsystem under the time trigger mechanism is established as follows:
wherein,representing xi i (θ i ) For theta i Partial derivative form of (a);as a function of the desired joint acceleration and the actual joint velocity; the optimal performance index function based on the time trigger mechanism is as follows:
by combining with the optimization thought, the optimal performance index functionSatisfying the Hamiltonian equation:
if the optimum performance index functionIs continuously differentiable and satisfies the above equation, the corresponding Hamiltonian presence solution of a nonlinear continuous system is used as an optimal control strategy +.>Expressed as:
bringing the optimal control strategy into the Hamiltonian, deriving:
step three, based on the event triggering mechanism principle, if and only if the triggering condition is violated, calculating and updating the tracking control law input of the modularized mechanical arm system; assume thatIs a group of monotone increasing sequences composed of trigger moments, which satisfies 0 < t 1 ＜t l+1 And there is +.f. for l ε {0,1,2, … }>Defining the sampling moment state as follows:
wherein,for t E [ t ] 1 ，t l+1 ) Sampling data information of states in time; to obtain a suitable event triggering condition, an interval function E is defined which represents the relation between the sampled state and the actual state information 1i (t) is:
according to the event trigger mechanism, the control law is updated following the trigger condition, that is, the state at this time is expressed as:
under an event triggering mechanism, in order to ensure the safe operation of the modularized mechanical arm, the tracking control law changes a discrete signal into a continuous signal through a zero-order retainer, and the expression is as follows:
at t E [ t ] 1 ，t l+1 ) In the moment, according to the optimal control strategy under the time trigger mechanism, defining the optimal control strategy of the event trigger mechanism as follows:
however, the process is not limited to the above-described process,discrete values sampled non-periodically, the control signal is changed from discrete signal to continuous signal by introducing zero-order retainer; combining Hamiltonian equation based on time trigger mechanism and event trigger based machineThe optimal control law is prepared, and the Hamiltonian equation based on the event triggering mechanism is obtained as follows:
for all statesThere is a designed tracking control law that is Lipschitz continuous and there is a given positive constant m li The following relationship is satisfied:
tracking control strategy based on event trigger mechanism, tracking error equation of sub-joint module system according to trigger time interval t 1 Will be theta i (x i ) Is replaced byThe tracking control strategy is updated according to the event trigger condition
Step four, in order to solve the Hamiltonian based on the event trigger mechanism, the strong learning ability of the neural network is utilized to approximate the performance index function (XI) i (θ i )：
Wherein,is ideal weight, ni is the number of hidden layer neurons, < ->Is an activation function, ε ei (θ i ) To evaluate neural network approximation residuals; performance index function based on event trigger mechanism +.>The partial derivative of (2) is expressed as:
wherein,and->The partial derivatives of the activation function and the evaluation neural network approximation residual are respectively; let->And->δ eid And g eid Is a positive constant; deducing inequalityIs true, where P ei For a given control parameter; thus, the optimal control law is approximated as:
the Hamiltonian obtained based on the event trigger mechanism is expressed as:
however, due to the ideal evaluation neural network weight W ei We have no thingThe method is directly known, so the evaluation neural network is approximated as:
wherein,is an approximation of the neural network weights; thus, an approximate event-triggered distributed tracking control strategy is obtained:
the approximate Hamiltonian is derived as:
wherein ε EHi Is an estimated residual of the approximate event-triggered Hamiltonian; next, an evaluation neural network weight estimation error is defined:in order to evaluate the estimated weights of the neural network +.>Tracked upper expected weight W ei Minimizing the objective function by using gradient descent algorithm>Adjusting the weight vector of the neural network>Its update strategy is designed:
wherein alpha is ei To evaluate the update law of network weight, designFinally, an approximate event-triggered distributed tracking control strategy is obtained:
according to the Lyapunov stability theorem and an event triggering mechanism, an event triggering condition which can stabilize the system is designed for the modularized mechanical arm joint module subsystem with n degrees of freedom, namely:
wherein lambda is min () And lambda (lambda) max () Respectively representing the minimum eigenvalue and the maximum eigenvalue of the matrix in the brackets; e (E) LEi Event triggering conditions for dispersing optimal tracking control are triggered for the proposed modular mechanical arm events.
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