CN115502986A - Multi-joint mechanical arm event drive control method based on state observer - Google Patents
Multi-joint mechanical arm event drive control method based on state observer Download PDFInfo
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Abstract
The invention relates to a multi-joint mechanical arm event-driven control method based on a state observer, which is characterized in that a multi-joint mechanical arm nonlinear dynamic model is established; adopting an RBF neural network to approximate unmodeled dynamics in a state equation, thereby constructing a state observer; designing a convergence track with a preset performance function and a nonlinear transformation limit tracking error; designing an instruction filter with a low-order compensation system according to an unconstrained tracking error and a reverse-thrust design method; constructing unmodeled dynamics of an RBF neural network approximation control loop to obtain an estimation model; and designing a mechanical arm event drive control law according to the estimation of the state observer, the estimation of the instruction filter and the estimation of the neural network. The invention realizes the convergence of the track tracking error with preset performance on the premise of not using mechanical arm model parameters and joint angular velocity sensors. In addition, the control method of the invention has simple structure and can effectively reduce the data transmission between the controller and the execution mechanism.
Description
Technical Field
The invention relates to the field of mechanical arm control, in particular to a multi-joint mechanical arm event-driven control method based on a state observer.
Background
With the push and development of industry 4.0 and china manufacturing 2025, the mechanical arm is gaining wide attention as an important link of the intelligent manufacturing industry. In order to ensure that the mechanical arm can be safely and reliably used for industrial production, higher requirements are put forward on the dynamic performance and steady-state precision of a mechanical arm control system. Articulated robotic arms are a non-linear, multivariable, strongly coupled mechanical system and present unmodeled dynamics that present challenges to the preset performance control of the robotic arm. In addition, due to space and cost constraints, multi-joint robotic arms are often difficult to equip with joint angular velocity sensors. In the prior art, the preset performance control of the single-joint mechanical arm based on the observer can be realized on the premise of utilizing the model parameters of the mechanical arm. However, the observer and the control method designed for the single-joint robot arm cannot be applied to the multi-joint robot arm, and it is difficult to obtain accurate model parameters in practice.
The traditional mechanical arm control method needs to take a time derivative for a virtual control law and then construct a neural network to approximate the unmodeled dynamics of a mechanical arm system. However, derivative terms of the virtual control laws can cause the number of dimensions of the excitation function vectors of the neural network to increase, thereby increasing the amount of computation. In addition, the conventional compensation system has more state variables, which causes the control method to have a complex structure and is not beneficial to practical application.
In a classical mechanical arm sampling control system, output data of a controller is transmitted to an actuating mechanism in real time. If the sampling frequency is too high, not only a great deal of communication resources are wasted, but also the execution mechanism is caused to act frequently, thereby reducing the service life of the execution mechanism. The event-driven control has the advantages that the controller updates the output data only when the triggering condition is met, the data transmission between the controller and the executing mechanism and the action times of the executing mechanism can be reduced, and the original control performance is ensured. In the prior art, on the premise of utilizing a joint angular velocity sensor, event-driven multi-joint mechanical arm track tracking control can be realized. However, on the premise of not using multi-joint mechanical arm model information and joint angular velocity sensors, it is a difficult point in the field of mechanical arm control to realize trajectory tracking event-driven control that ensures preset performance.
Disclosure of Invention
The invention provides a multi-joint mechanical arm event-driven control method based on a state observer. The method aims to solve the problems that the track tracking error of a multi-joint mechanical arm cannot be converged in a preset performance, communication resources of a control system are greatly wasted and the like on the premise that the existing control method does not depend on mechanical arm model parameters and joint angular velocity sensors.
In order to achieve the purpose, the invention adopts the following technical scheme that:
the multi-joint mechanical arm event drive control method based on the state observer comprises the following steps:
and 6, designing an event-driven control law of the mechanical arm according to the estimated joint angular velocity and the output of the instruction filter by combining an unmodeled dynamic estimation model and an event-driven method, so that the actual track tracks the expected track with preset performance.
Further, the nonlinear dynamic model of the multi-joint mechanical arm in the step 1 is as follows:
wherein,a vector of the angular position of the joint is represented,the angular velocity vector of the joint is represented,represents the angular acceleration vector of the joint,the control torque provided for the motor is,in order to define the inertia matrix in a symmetrical positive way,is a matrix of centrifugal forces and coriolis forces,is a gravity vector.
Further, the state equation of the multi-joint mechanical arm in the step 1 is as follows:
Further, the step 2 of approximating unmodeled dynamics in the state equation by using the RBF neural network includes:
wherein,a vector of the excitation function is represented,represents an approximation error, and satisfies,Is a normal number which is a positive number,is an ideal weight matrix and the weight matrix is,is a vector of the gaussian basis function,andcan be expressed as
Wherein,the number of the nodes of the RBF neural network,is as followsThe center vector of each of the nodes is,is as followsThe gaussian-based width of an individual node,representing an exponential function.
Further, in step 2, the state observer is constructed as follows:
wherein,,,,,for design parameters, and for positive numbers,to representThe estimated amount of (a) is,indicating angular velocity of jointThe estimated amount of (a) is,andrespectively representAndthe estimated amount of (a) is,is the intermediate auxiliary variable.
Further, the unconstrained tracking error in step 3 is
Wherein,,for the purpose of an unconstrained tracking error,andrespectively setting initial values of a preset performance function and a tracking error;
wherein, the tracking error is:
wherein,for the desired joint angle position vector to be,in order to provide a tracking error for the mechanical arm track,represents a joint angle position vector;
the preset performance function is:
wherein,,representation for limiting tracking errorThe performance of the device is preset according to the performance function,,,is a design parameter, and,andis a positive number, and the number of the positive number,representing an exponential function. Parameter(s)For designing convergence time, parameters, of tracking errorTo design the convergence accuracy of the tracking error.
If the performance function is presetThen, thenNamely, the control method of the present invention degenerates to the conventional non-default performance control method.
Further, in step 4, the instruction filter is:
wherein,in order to design the parameters of the device,is a virtual control law with the expression of
Wherein,representing design parameters and being a positive definite matrix, unconstrained tracking error vector,,To for,,,To account for the tracking error of the compensated signal, the expression is
Wherein,to compensate the signal, it is generated by a low-order compensation system; constructing a low order compensation system of
Wherein,the design parameters are represented by a number of parameters,,representing a symbolic function.
Further, the estimation model of unmodeled dynamics in step 5 is
Wherein,for the unmodeled dynamics of the control loop,a vector of the excitation function is represented,represents an approximation error, and satisfies,Is a normal number which is a positive number,is an ideal weight matrix of the weight values,is a vector of the gaussian basis function,andcan be expressed as
Wherein,the number of the nodes of the RBF neural network,is as followsThe center vector of each of the nodes is,is as followsThe gaussian-based width of an individual node,representing an exponential function.
If there is no instruction filter, the virtual control law is neededThe time derivative is taken as a function of time,is a variable quantity、、、、、、、The function of (a), is quite complex. The unmodeled dynamics of the control loop isSo that it is necessary to increase neural network excitationThe dimension of the function vector can effectively estimate the unmodeled dynamics of the control loop, thereby increasing the calculation amount of the control method.
Further, in step 6, the event-driven control law of the mechanical arm is as follows:
wherein,,is a positive integer and is a non-zero integer,、andto design parameters and satisfy,Is a function of the hyperbolic tangent,,,、、andare respectively as、、Andto (1) aA component;
Wherein,representing design parameters, and is a positive definite matrix,andin order to design the parameters of the device,,andrespectively representAndis measured.
Further, the stability proving method of the control method comprises the following steps:
defining an estimated error variable,,Whereinis in a state(ii) an estimate of (d); for estimation error variableAndrespectively taking time derivatives of
Wherein,,according to the Gaussian base functionIs characterized by obtainingWherein, in the process,is a normal number, further obtained
According to the Young's inequality
According to the inequality
,(ii) a From the above equation, an error variable is estimated,,Is bounded, i.e. stores a constant,,So that,,It is true that, among other things,;
constructing the Lyapunov functionTo is aligned withTaking the time derivative and taking into account the compensating system
Wherein the error variableTaking into account the parameters of the mechanical armAndis obliquely symmetrical with respect toTaking the time derivative to obtain
Wherein the error variable,Is a time-varying function, satisfies,Is a normal number of the blood vessel which is,is a bounded function vector; according to the Young's inequality
Due to the fact thatFor positive definite diagonal matrix and for instruction filter, there isSo thatIf true; thus by selectionTo obtain
According to the formula, the compound has the advantages of,is asymptotically stable, i.e. whenWhen the utility model is used, the water is discharged,(ii) a Constructing the Lyapunov functionTo, forTaking the time derivative to obtain
,, (ii) a According to the formula, the compound has the advantages of,、、is bounded; according toIn the knowledge that,is bounded and tracking errorAre subject to preset performance requirements, i.e.(ii) a Because of the desired trajectoryIs bounded, thereforeAndis bounded, which means thatAndis bounded; further obtain the、、、、Andis bounded and therefore the closed loop system is stable;
From the above derivation, there are normal numbersSo thatIf true; because ofAndso that adjacent trigger time intervals(ii) a Therefore, the event-driven control law designed by the invention is reasonable, namely, the Seno behavior is avoided.
Compared with the prior art, the invention has the following advantages and beneficial effects:
1. under the condition of not utilizing parameters of a mechanical arm model, constructing an unmodeled dynamic state in an RBF neural network approximation state equation, and realizing the estimation of the multi-joint angular velocity;
2. according to the invention, on the premise of not depending on mechanical arm model parameters and joint angular velocity sensors, the convergence of a track tracking error according to preset performance is realized, and the dynamic performance and the steady-state precision of a control system are improved;
3. the invention designs the instruction filter with a low-order compensation system to process the time derivative of the virtual control law, and reduces the dimension of the excitation function vector of the neural network, so that the control method has a simple structure and is convenient for practical application;
4. the invention adopts an event trigger control method, can effectively reduce the data transmission between the controller and the executing mechanism, and ensures the preset control performance.
Based on the reasons, the invention can be widely popularized in the field of mechanical arm control.
Drawings
FIG. 1 is a flow chart of a control method of the present invention;
FIG. 2 is a graph comparing the tracking effect of the angular position of the joint 1 after different control methods are adopted;
FIG. 3 is a graph comparing the tracking effect of the angular position of the joint 2 after different control methods are adopted;
FIG. 4 is a graph comparing tracking errors of the angular position of the joint 1 after different control methods are adopted;
FIG. 5 is a graph comparing tracking errors of angular positions of joints 2 after different control methods are used;
FIG. 6 is a diagram showing the angular velocity and the estimation effect of the joint 1 according to the control method of the present invention;
FIG. 7 is a diagram showing the angular velocity of the joint 2 and the effect of estimation thereof according to the control method of the present invention;
FIG. 8 is a control moment diagram of the joint 1 according to the control method of the present invention;
FIG. 9 is a control torque diagram of the joint 2 according to the control method of the present invention;
FIG. 10 shows a neural network weight Fan Shutu according to the control method of the present invention;
FIG. 11 shows the neural network weights Fan Shutu according to the control method of the present invention;
FIG. 12 is a graph showing the tracking error of the angular position of the joint 1 in comparison with each other;
fig. 13 is a graph comparing tracking errors in the angular position of the joint 2 in different cases.
Detailed Description
The invention is described in more detail below with reference to the accompanying drawings.
Aiming at the problem of tracking the track of a multi-joint mechanical arm based on a state observer and event driving, an RBF neural network is adopted to approximate unmodeled dynamic state in a state equation, so that the state observer is constructed to estimate the angular velocity of a joint, a convergence track with a preset performance function and nonlinear transformation limiting tracking error is designed, an instruction filter with a low-order compensation system is designed to process the time derivative of a virtual control law according to unconstrained tracking error and a reverse-pushing design method, the excitation function vector of the neural network is simplified, the control method is simple in structure and convenient to actually apply, unmodeled dynamic state of an RBF neural network approximation control loop is constructed to obtain an estimation model of unmodeled dynamic state, and the event driving control law of the mechanical arm is designed by combining the estimation model of unmodeled dynamic state and the event driving method according to the estimated angular velocity of the joint and the output of the instruction filter, so that the actual track tracks the expected track with the preset performance.
As shown in fig. 1, the invention provides a multi-joint mechanical arm event-driven control method based on a state observer, which comprises the following steps:
the established nonlinear dynamic model of the multi-joint mechanical arm with unmodeled dynamics is as follows:
wherein,a vector of the angular position of the joint is represented,the angular velocity vector of the joint is represented,represents the angular acceleration vector of the joint,the control torque provided for the motor is,in order to define the inertia matrix in a symmetrical positive way,is a matrix of centrifugal forces and coriolis forces,is a gravity vector.
The state equation is:
the unmodeled dynamics in the RBF neural network approximation state equation is:
wherein,a vector of the excitation function is represented,represents an approximation error, and satisfies,Is a normal number which is a positive number,is an ideal weight matrix and the weight matrix is,is a vector of the gaussian basis function,andcan be expressed as
Wherein,the number of the nodes of the RBF neural network,is as followsThe center vector of each of the nodes is,is as followsThe gaussian base width of each node is,representing an exponential function.
The state observer is constructed as follows:
wherein,,,,,for design parameters, and for positive numbers,to representThe estimated amount of (a) is,indicating angular velocity of jointThe estimated amount of (a) is,andrespectively representAndthe estimated amount of (a) is,is the intermediate auxiliary variable.
the tracking error is:
wherein,for the desired joint angle position vector to be,in order to provide a tracking error for the mechanical arm track,represents a joint angle position vector;
the preset performance function is:
wherein,,representation for limiting tracking errorThe function of the preset performance of the system,,,is a design parameter, and,andis a positive number of the bits,representing exponential functions, parametersFor designing convergence time, parameters, of tracking errorTo design the convergence accuracy of the tracking error.
The unconstrained tracking error obtained by the nonlinear transformation is:
wherein,,for the purpose of an unconstrained tracking error,andrespectively, a preset performance function and an initial value of the tracking error.
If the performance function is presetThen, thenNamely, the control method of the present invention degenerates to the conventional non-default performance control method.
the instruction filter is:
wherein,in order to design the parameters of the device,is a virtual control law with the expression of
Wherein,representing design parameters and being a positive definite matrix, unconstrained tracking error vector,,To a,,,To account for the tracking error of the compensated signal, the expression is
Wherein,to compensate the signal, it is generated by a low-order compensation system; constructing a low order compensation system of
Wherein,the design parameters are represented by a number of parameters,,representing a symbolic function.
the unmodeled dynamics of the control loop are:
constructing RBF neural network to obtain unmodeled dynamic estimation model
Wherein,a vector of the excitation function is represented,represents an approximation error, and satisfies,Is a normal number of the cells, and,is an ideal weight matrix and the weight matrix is,is a vector of the gaussian basis function,andcan be expressed as
Wherein,the number of the nodes of the RBF neural network,is as followsThe center vector of each of the nodes is,is as followsThe gaussian-based width of an individual node,representing an exponential function.
If there is no instruction filter, the virtual control law is neededThe time derivative is taken as a function of time,is a variable quantity、、、、、、、Is very complex. The unmodeled dynamics of the control loop isThe number of dimensions of the excitation function vector of the neural network needs to be increased to effectively estimate the unmodeled dynamics of the control loop, so that the calculation amount of the control method is increased.
And 6, designing an event-driven control law of the mechanical arm according to the estimated joint angular velocity and the output of the instruction filter by combining an unmodeled dynamic estimation model and an event-driven method, so that the actual track tracks the expected track with preset performance.
The event-driven control law of the mechanical arm is as follows:
wherein,,is a positive integer and is a non-zero integer,、andto design parameters and satisfy,Is a function of the hyperbolic tangent,,,、、andare respectively as、、Andto (1)A component;
Wherein,representing design parameters, and is a positive definite matrix,andin order to design the parameters of the device,,andrespectively representAndis measured.
The stability proving method of the multi-joint mechanical arm event-driven control method based on the state observer comprises the following steps:
defining an estimated error variable,,Wherein, in the process,is in a state(ii) an estimate of (d); for estimation error variableAndrespectively taking time derivatives of
Wherein,,according to the characteristics of a Gaussian base functionWhereinis a normal number, further obtained
According to the Young inequality
According to the inequality
,(ii) a From the above equation, an error variable is estimated,,Is bounded, i.e. stores a constant,,So that,,It is true that, among other things,;
constructing Lyapunov functionsTo, forTaking the time derivative and taking into account the compensating system
Wherein the error variableTaking into account the parameters of the mechanical armAndis obliquely symmetrical with respect toTaking the time derivative to obtain
Wherein the error variable,Is a time-varying function, satisfies,Is a normal number, and is,is a bounded function vector; according to the Young's inequality
Due to the fact thatFor positive definite diagonal matrix and for instruction filter, there isSo thatIf true; thus by selectionTo obtain
According to the formula, the compound has the advantages of,is asymptotically stable, i.e. whenWhen the temperature of the water is higher than the set temperature,(ii) a Constructing Lyapunov functionsTo, forTaking the time derivative to obtain
,, (ii) a According to the formula, the compound has the advantages of,、、is bounded; according toIn the knowledge that,is bounded and tracking errorAre subject to predetermined performance requirements, i.e.(ii) a Because of the desired trajectoryIs bounded, thereforeAndis bounded, which means thatAndis bounded; further obtain the、、、、Andis bounded and therefore the closed loop system is stable;
From the above derivation, there are normal numbersSo thatIf true; because of the fact thatAndso that adjacent trigger time intervals(ii) a Therefore, the event-driven control law designed by the invention is reasonable, namely, the Seno behavior is avoided.
The behaviour of the sesame: in event-driven control, the controller is triggered an infinite number of times within a finite time. If the event driven controller is designed not to exclude the Chinoy behavior, the event driven controller is not effective and cannot be practically applied. The event-driven control method proves that the adjacent trigger time intervals of the controllers are proved by theoryWhereinis a normal number, i.e. the controller is not triggered an unlimited number of times in a limited time, so the event driven control law designed by the present invention is reasonable, i.e. does not have the action of sesame.
A simulation experiment is carried out on the designed multi-joint mechanical arm event-driven control method based on the state observer under the virtual environment, so that the feasibility of the method is verified. In a simulation experiment, the nonlinear dynamic model of the double-joint mechanical arm is as follows:
wherein,,andthe angular positions of the joint 1 and the joint 2 respectively,,andthe angular velocities of the joint 1 and the joint 2 respectively,,andcontrol moments, inertia matrices, of joints 1 and 2, respectivelyCentrifugal and coriolis force matricesAnd the gravity vectorRespectively as follows:
The preset performance function of the tracking error is set by the user according to the actual situation and, in this embodiment,,,。
to further illustrate the effectiveness of the control method designed by the present invention, a comparative experiment was conducted with the conventional control method (non-default performance control method). Order to(other control parameters are unchanged), the control method designed by the invention is changed into the traditional control method.
Fig. 2 and 3 are respectively a comparison graph of the track tracking of the angular positions of the joint 1 and the joint 2, and the control method of the present invention and the conventional control method can realize the track tracking of the angular positions of the joint, and as can be seen from fig. 2, for the angular position of the joint 1, the control method of the present invention coincides with the expected track at about 4.5s, and the conventional control method coincides with the expected track at about 6.5 s; as can be seen from fig. 3, for the angular position of the joint 2, the control method of the present invention coincides with the desired trajectory in about 4s, and the conventional control method coincides with the desired trajectory in about 5.5s, so that the control method of the present invention has a fast trajectory tracking speed, i.e., has good dynamic performance.
Fig. 4 and fig. 5 are comparison graphs of track tracking errors of angular positions of the joint 1 and the joint 2, respectively, and it can be seen from the graphs that, compared with the conventional control method, the convergence speed of the tracking error of the control method of the present invention is faster, and the convergence curve of the tracking error is within the preset performance function, however, the tracking error curve corresponding to the conventional control method cannot meet the requirement of the preset performance function.
Fig. 6 and 7 are diagrams of angular velocities of the joint 1 and the joint 2 and their estimation effects, respectively, and it can be seen from the diagrams that the observer designed in the present invention can realize the estimation of the angular velocity of the joint without depending on the parameters of the robot arm model.
Fig. 8 and 9 are control moment diagrams of the control method of the present invention, and it can be seen from the diagrams that the control signal generated by the event-driven control method of the robot arm according to the present invention is a piecewise constant, i.e. the signal transmission from the robot arm controller to the actuator is a piecewise constant, which effectively reduces data transmission and improves the resource utilization rate of the control system.
FIG. 10 and FIG. 11 are the weight norm of the neural network according to the control method of the present inventionAndthe weight norm of the neural network can be known from the graphAndthe method is bounded, namely the neural network weight value adjusting method designed by the invention is reasonable.
To further illustrate the effectiveness of the control method of the present invention, the predetermined performance function is set in two cases. First parameter setting case:,,. Second parameter setting case:,,. As can be seen from the parameter setting cases, the joint angle position tracking error of the second parameter setting case will be converged within 3 seconds, and has a high steady-state accuracy.
Fig. 12 and 13 are graphs showing the comparison of the tracking errors of the angular positions of the joint 1 and the joint 2 in different cases, and it can be seen that the convergence rate of the tracking error is high in the case of the second parameter setting, and the tracking error converges to around zero in about 2.5 seconds.
In order to quantitatively compare the control performance of the setting conditions of the two parameters, the invention adopts integral time absolute error and integral absolute errorRespectively evaluating the dynamic performance and the steady-state precision of the error signal, wherein the expression of the absolute error of the integral time isThe expression of the integral absolute error isWherein,as a matter of time, the time is,is the tracking error. The performance indexes of different parameter setting conditions are listed in the table I, and the fact that the second parameter setting condition has smaller integral time absolute error and integral absolute error can be found from the table I, namely, the error signal corresponding to the second parameter setting condition has better dynamic performance and steady-state precision, so that the control method can realize the convergence of the tracking error with preset performance.
table-Performance index for different parameter settings
The simulation experiment result shows that the method realizes the convergence of the track tracking error with the preset performance on the premise of not depending on the parameters of the mechanical arm model and the joint angular velocity sensor, and has better dynamic performance and steady-state precision. In addition, the control method of the invention has simple structure, can effectively reduce the data transmission between the controller and the execution mechanism, and improves the resource utilization rate of the control system.
It should be understood that the detailed description and specific examples, while indicating preferred embodiments of the invention, are given by way of illustration only, not limitation, and it will be understood by those skilled in the art that various changes and modifications may be made therein without departing from the spirit and scope of the invention; as long as the use requirements are met, the method is within the protection scope of the invention.
Claims (10)
1. A multi-joint mechanical arm event drive control method based on a state observer is characterized by comprising the following steps:
step 1, establishing a nonlinear dynamic model of the multi-joint mechanical arm with unmodeled dynamics, and converting the nonlinear dynamic model of the multi-joint mechanical arm into a state equation;
step 2, adopting an RBF neural network to approximate unmodeled dynamics in a state equation, thereby constructing a state observer to estimate the angular velocity of the joint;
step 3, aiming at the tracking error generated by the nonlinear dynamic model of the multi-joint mechanical arm, designing a convergence track with a preset performance function and nonlinear transformation limiting the tracking error to obtain an unconstrained tracking error;
step 4, designing an instruction filter with a low-order compensation system according to an unconstrained tracking error and a reverse-thrust design method;
step 5, constructing unmodeled dynamics of the RBF neural network approximation control loop to obtain an estimation model of the unmodeled dynamics;
and 6, designing an event-driven control law of the mechanical arm according to the estimated joint angular velocity and the output of the instruction filter by combining an unmodeled dynamic estimation model and an event-driven method, so that the actual track tracks the expected track with preset performance.
2. The state observer-based multi-joint mechanical arm event-driven control method according to claim 1, characterized in that: the nonlinear dynamic model of the multi-joint mechanical arm in the step 1 is as follows:
wherein,a vector of the angular position of the joint is represented,the angular velocity vector of the joint is represented,represents the angular acceleration vector of the joint,the control torque provided for the motor is,in order to define the inertia matrix in a symmetrical positive way,is a matrix of centrifugal forces and coriolis forces,is a gravity vector.
4. The state observer-based multi-joint mechanical arm event-driven control method according to claim 1, characterized in that: in the step 2, the unmodeled dynamics in the RBF neural network approximation state equation is adopted as follows:
wherein,a vector of the excitation function is represented,represents an approximation error, and satisfies,Is a normal number which is a positive number,is an ideal weight matrix of the weight values,is a vector of the gaussian basis function,andcan be expressed as
5. The state observer-based multi-joint mechanical arm event-driven control method according to claim 1, characterized in that: in step 2, a state observer is constructed as follows:
6. The state observer-based multi-joint mechanical arm event-driven control method according to claim 1, characterized in that: unconstrained tracking error in step 3 is
Wherein,,for the purpose of an unconstrained tracking error,andrespectively setting initial values of a preset performance function and a tracking error;
wherein, the tracking error is:
wherein,for the desired joint angle position vector to be,in order to provide a tracking error for the mechanical arm track,represents a joint angle position vector;
the preset performance function is:
7. The multi-joint mechanical arm event-driven control method based on the state observer as claimed in claim 1, wherein: the instruction filter in step 4 is:
wherein,in order to design the parameters of the device,is a virtual control law, and the expression is
Wherein,representing design parameters and being a positive definite matrix, unconstrained tracking error vector,,To a,,,To account for the tracking error of the compensated signal, the expression is
Wherein,to compensate the signal, it is generated by a low-order compensation system; constructing a low order compensation system of
8. The state observer-based multi-joint mechanical arm event-driven control method according to claim 1, characterized in that: the estimation model of unmodeled dynamics in step 5 is
Wherein,for the unmodeled dynamics of the control loop,a vector of the excitation function is represented,represents an approximation error, and satisfies,Is a normal number which is a positive number,is an ideal weight matrix and the weight matrix is,is a vector of the gaussian basis function,andcan be expressed as
9. The state observer-based multi-joint mechanical arm event-driven control method according to claim 1, characterized in that: the mechanical arm event driving control law in the step 6 is as follows:
wherein,,is a positive integer and is a non-zero integer,、andto design parameters and satisfy,Is a function of the hyperbolic tangent,,,、、andare respectively as、、Andto (1) aA component;
10. The state observer-based multi-joint mechanical arm event-driven control method according to claim 1, characterized in that: the stability proving method of the control method comprises the following steps:
defining an estimated error variable,,Whereinis in a state(ii) an estimate of (d); for the estimated error variableAndrespectively taking time derivatives of
Wherein,,according to the characteristics of a Gaussian base functionWhereinis a normal number, further obtained
According to the Young inequality
According to the above inequality
,(ii) a From the above equation, an error variable is estimated,,Is bounded, i.e. stores a normal number,,So that,,It is true that, among other things,;
constructing Lyapunov functionsTo, forTaking the time derivative and taking into account the compensating system
Wherein the error variableTaking into account the parameters of the mechanical armAndoblique symmetry of (2), toTaking the time derivative to obtain
Wherein the error variable,Is a time-varying function, satisfies,Is a normal number, and is,is a bounded function vector; according to the Young's inequality
Due to the fact thatFor positive definite diagonal matrixAnd for the instruction filter, there isSo thatIf true; thus by selectionTo obtain
According to the formula, the compound has the advantages of,is asymptotically stable, i.e. whenWhen the temperature of the water is higher than the set temperature,(ii) a Constructing the Lyapunov functionTo, forTaking the time derivative to obtain
,, (ii) a According to the formula, the compound has the advantages of,、、is bounded; according toIn the knowledge that,is bounded and tracking errorAre subject to preset performance requirements, i.e.(ii) a Because of the desired trajectoryIs bounded, thereforeAndis bounded, which means thatAndis bounded; further obtain the、、、、Andis bounded and therefore the closed loop system is stable;
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