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 PDF

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CN115502986A
CN115502986A CN202211421097.4A CN202211421097A CN115502986A CN 115502986 A CN115502986 A CN 115502986A CN 202211421097 A CN202211421097 A CN 202211421097A CN 115502986 A CN115502986 A CN 115502986A
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mechanical arm
joint
function
tracking error
control method
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CN115502986B (en
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杨迪
邹臣禧
刘伟军
赵海超
张恒
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Shenyang University of Technology
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    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
    • B25J9/16—Programme controls
    • B25J9/1602—Programme controls characterised by the control system, structure, architecture
    • B25J9/161—Hardware, e.g. neural networks, fuzzy logic, interfaces, processor
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
    • B25J9/16—Programme controls
    • B25J9/1602—Programme controls characterised by the control system, structure, architecture
    • B25J9/1605—Simulation of manipulator lay-out, design, modelling of manipulator

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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

Multi-joint mechanical arm event drive control method based on state observer
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:
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.
Further, the nonlinear dynamic model of the multi-joint mechanical arm in the step 1 is as follows:
Figure 213205DEST_PATH_IMAGE001
wherein,
Figure DEST_PATH_IMAGE002
a vector of the angular position of the joint is represented,
Figure 214659DEST_PATH_IMAGE003
the angular velocity vector of the joint is represented,
Figure DEST_PATH_IMAGE004
represents the angular acceleration vector of the joint,
Figure 882401DEST_PATH_IMAGE005
the control torque provided for the motor is,
Figure DEST_PATH_IMAGE006
in order to define the inertia matrix in a symmetrical positive way,
Figure 957804DEST_PATH_IMAGE007
is a matrix of centrifugal forces and coriolis forces,
Figure DEST_PATH_IMAGE008
is a gravity vector.
Further, the state equation of the multi-joint mechanical arm in the step 1 is as follows:
Figure 685589DEST_PATH_IMAGE009
wherein,
Figure DEST_PATH_IMAGE010
，
Figure 174339DEST_PATH_IMAGE011
，
Figure DEST_PATH_IMAGE012
representing unmodeled dynamics by the expression
Figure 580525DEST_PATH_IMAGE013
。
Further, the step 2 of approximating unmodeled dynamics in the state equation by using the RBF neural network includes:
Figure DEST_PATH_IMAGE014
wherein,
Figure 572752DEST_PATH_IMAGE015
a vector of the excitation function is represented,
Figure DEST_PATH_IMAGE016
represents an approximation error, and satisfies
Figure 471438DEST_PATH_IMAGE017
，
Figure DEST_PATH_IMAGE018
Is a normal number which is a positive number,
Figure 181905DEST_PATH_IMAGE019
is an ideal weight matrix and the weight matrix is,
Figure DEST_PATH_IMAGE020
is a vector of the gaussian basis function,
Figure 191449DEST_PATH_IMAGE019
and
Figure 38182DEST_PATH_IMAGE020
can be expressed as
Figure 842190DEST_PATH_IMAGE021
Wherein for
Figure DEST_PATH_IMAGE022
，
Figure 39954DEST_PATH_IMAGE023
，
Figure DEST_PATH_IMAGE024
The expression of the Gaussian basis function is
Figure 587610DEST_PATH_IMAGE025
Wherein,
Figure DEST_PATH_IMAGE026
the number of the nodes of the RBF neural network,
Figure 554429DEST_PATH_IMAGE027
is as follows
Figure DEST_PATH_IMAGE028
The center vector of each of the nodes is,
Figure 794917DEST_PATH_IMAGE029
is as follows
Figure 214397DEST_PATH_IMAGE028
The gaussian-based width of an individual node,
Figure DEST_PATH_IMAGE030
representing an exponential function.
Further, in step 2, the state observer is constructed as follows:
Figure 565744DEST_PATH_IMAGE031
wherein,
Figure DEST_PATH_IMAGE032
，
Figure 387069DEST_PATH_IMAGE033
，
Figure DEST_PATH_IMAGE034
，
Figure 64038DEST_PATH_IMAGE035
，
Figure DEST_PATH_IMAGE036
for design parameters, and for positive numbers,
Figure 985463DEST_PATH_IMAGE037
to represent
Figure DEST_PATH_IMAGE038
The estimated amount of (a) is,
Figure 874922DEST_PATH_IMAGE039
indicating angular velocity of joint
Figure DEST_PATH_IMAGE040
The estimated amount of (a) is,
Figure 816333DEST_PATH_IMAGE041
and
Figure DEST_PATH_IMAGE042
respectively represent
Figure 336307DEST_PATH_IMAGE043
And
Figure DEST_PATH_IMAGE044
the estimated amount of (a) is,
Figure 730379DEST_PATH_IMAGE045
is the intermediate auxiliary variable.
Further, the unconstrained tracking error in step 3 is
Figure DEST_PATH_IMAGE046
Wherein,
Figure 423529DEST_PATH_IMAGE047
，
Figure DEST_PATH_IMAGE048
for the purpose of an unconstrained tracking error,
Figure 219446DEST_PATH_IMAGE049
and
Figure DEST_PATH_IMAGE050
respectively setting initial values of a preset performance function and a tracking error;
wherein, the tracking error is:
Figure 238218DEST_PATH_IMAGE051
wherein,
Figure DEST_PATH_IMAGE052
for the desired joint angle position vector to be,
Figure 119586DEST_PATH_IMAGE053
in order to provide a tracking error for the mechanical arm track,
Figure DEST_PATH_IMAGE054
represents a joint angle position vector;
the preset performance function is:
Figure 350847DEST_PATH_IMAGE055
wherein,
Figure DEST_PATH_IMAGE056
，
Figure 204534DEST_PATH_IMAGE057
representation for limiting tracking error
Figure DEST_PATH_IMAGE058
The performance of the device is preset according to the performance function,
Figure 128628DEST_PATH_IMAGE059
，
Figure DEST_PATH_IMAGE060
，
Figure 497292DEST_PATH_IMAGE061
is a design parameter, and
Figure DEST_PATH_IMAGE062
，
Figure 797823DEST_PATH_IMAGE060
and
Figure 34245DEST_PATH_IMAGE063
is a positive number, and the number of the positive number,
Figure DEST_PATH_IMAGE064
representing an exponential function. Parameter(s)
Figure 129240DEST_PATH_IMAGE065
For designing convergence time, parameters, of tracking error
Figure 781938DEST_PATH_IMAGE060
To design the convergence accuracy of the tracking error.
If the performance function is preset
Figure DEST_PATH_IMAGE066
Then, then
Figure 355002DEST_PATH_IMAGE067
Namely, the control method of the present invention degenerates to the conventional non-default performance control method.
Further, in step 4, the instruction filter is:
Figure DEST_PATH_IMAGE068
wherein,
Figure 183281DEST_PATH_IMAGE069
in order to design the parameters of the device,
Figure DEST_PATH_IMAGE070
is a virtual control law with the expression of
Figure 449177DEST_PATH_IMAGE071
Wherein,
Figure DEST_PATH_IMAGE072
representing design parameters and being a positive definite matrix, unconstrained tracking error vector
Figure 792434DEST_PATH_IMAGE073
，
Figure DEST_PATH_IMAGE074
，
Figure 434768DEST_PATH_IMAGE075
To for
Figure DEST_PATH_IMAGE076
，
Figure 914291DEST_PATH_IMAGE077
，
Figure DEST_PATH_IMAGE078
，
Figure 351088DEST_PATH_IMAGE079
To account for the tracking error of the compensated signal, the expression is
Figure DEST_PATH_IMAGE080
Wherein,
Figure 119324DEST_PATH_IMAGE081
to compensate the signal, it is generated by a low-order compensation system; constructing a low order compensation system of
Figure DEST_PATH_IMAGE082
Wherein,
Figure 34190DEST_PATH_IMAGE083
the design parameters are represented by a number of parameters,
Figure DEST_PATH_IMAGE084
，
Figure 633799DEST_PATH_IMAGE085
representing a symbolic function.
Further, the estimation model of unmodeled dynamics in step 5 is
Figure DEST_PATH_IMAGE086
Wherein,
Figure 507077DEST_PATH_IMAGE087
for the unmodeled dynamics of the control loop,
Figure DEST_PATH_IMAGE088
a vector of the excitation function is represented,
Figure 559347DEST_PATH_IMAGE089
represents an approximation error, and satisfies
Figure DEST_PATH_IMAGE090
，
Figure 484096DEST_PATH_IMAGE091
Is a normal number which is a positive number,
Figure DEST_PATH_IMAGE092
is an ideal weight matrix of the weight values,
Figure 938211DEST_PATH_IMAGE093
is a vector of the gaussian basis function,
Figure DEST_PATH_IMAGE094
and
Figure 716811DEST_PATH_IMAGE093
can be expressed as
Figure 256377DEST_PATH_IMAGE095
Wherein, for
Figure DEST_PATH_IMAGE096
，
Figure 778625DEST_PATH_IMAGE097
，
Figure DEST_PATH_IMAGE098
The expression of the Gaussian base function is
Figure 87247DEST_PATH_IMAGE099
Wherein,
Figure DEST_PATH_IMAGE100
the number of the nodes of the RBF neural network,
Figure 36748DEST_PATH_IMAGE101
is as follows
Figure DEST_PATH_IMAGE102
The center vector of each of the nodes is,
Figure 63610DEST_PATH_IMAGE103
is as follows
Figure 123970DEST_PATH_IMAGE102
The gaussian-based width of an individual node,
Figure DEST_PATH_IMAGE104
representing an exponential function.
If there is no instruction filter, the virtual control law is needed
Figure 287098DEST_PATH_IMAGE105
The time derivative is taken as a function of time,
Figure DEST_PATH_IMAGE106
is a variable quantity
Figure 407501DEST_PATH_IMAGE107
、
Figure DEST_PATH_IMAGE108
、
Figure 921659DEST_PATH_IMAGE109
、
Figure DEST_PATH_IMAGE110
、
Figure 51289DEST_PATH_IMAGE111
、
Figure DEST_PATH_IMAGE112
、
Figure 68923DEST_PATH_IMAGE113
、
Figure DEST_PATH_IMAGE114
The function of (a), is quite complex. The unmodeled dynamics of the control loop is
Figure 625807DEST_PATH_IMAGE115
So 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:
Figure DEST_PATH_IMAGE116
wherein,
Figure 358752DEST_PATH_IMAGE117
，
Figure DEST_PATH_IMAGE118
is a positive integer and is a non-zero integer,
Figure 964177DEST_PATH_IMAGE119
、
Figure DEST_PATH_IMAGE120
and
Figure 101897DEST_PATH_IMAGE121
to design parameters and satisfy
Figure DEST_PATH_IMAGE122
，
Figure 829682DEST_PATH_IMAGE123
Is a function of the hyperbolic tangent,
Figure DEST_PATH_IMAGE124
，
Figure 787273DEST_PATH_IMAGE125
，
Figure DEST_PATH_IMAGE126
、
Figure 258706DEST_PATH_IMAGE127
、
Figure DEST_PATH_IMAGE128
and
Figure 516512DEST_PATH_IMAGE129
are respectively as
Figure DEST_PATH_IMAGE130
、
Figure 149619DEST_PATH_IMAGE131
、
Figure DEST_PATH_IMAGE132
And
Figure 860086DEST_PATH_IMAGE133
to (1) a
Figure DEST_PATH_IMAGE134
A component;
law of virtual control
Figure 72892DEST_PATH_IMAGE135
Is expressed as
Figure DEST_PATH_IMAGE136
Wherein,
Figure 919625DEST_PATH_IMAGE137
representing design parameters, and is a positive definite matrix,
Figure DEST_PATH_IMAGE138
and
Figure 989213DEST_PATH_IMAGE139
in order to design the parameters of the device,
Figure DEST_PATH_IMAGE140
，
Figure 921397DEST_PATH_IMAGE141
and
Figure DEST_PATH_IMAGE142
respectively represent
Figure 211DEST_PATH_IMAGE143
And
Figure DEST_PATH_IMAGE144
is measured.
Further, the stability proving method of the control method comprises the following steps:
defining an estimated error variable
Figure 958241DEST_PATH_IMAGE145
，
Figure DEST_PATH_IMAGE146
，
Figure 933150DEST_PATH_IMAGE147
Wherein
Figure DEST_PATH_IMAGE148
is in a state
Figure 821472DEST_PATH_IMAGE149
(ii) an estimate of (d); for estimation error variable
Figure DEST_PATH_IMAGE150
And
Figure 172819DEST_PATH_IMAGE151
respectively taking time derivatives of
Figure DEST_PATH_IMAGE152
Wherein,
Figure 994144DEST_PATH_IMAGE153
，
Figure DEST_PATH_IMAGE154
according to the Gaussian base functionIs characterized by obtaining
Figure 405534DEST_PATH_IMAGE155
Wherein, in the process,
Figure DEST_PATH_IMAGE156
is a normal number, further obtained
Figure 312310DEST_PATH_IMAGE157
The following Lyapunov function is constructed
Figure DEST_PATH_IMAGE158
To analyze the stability of the observer
Figure 467348DEST_PATH_IMAGE159
To Lyapunov function
Figure 877601DEST_PATH_IMAGE158
Taking the time derivative to obtain
Figure DEST_PATH_IMAGE160
According to the Young's inequality
Figure 459892DEST_PATH_IMAGE161
According to the inequality
Figure DEST_PATH_IMAGE162
Wherein,
Figure 119543DEST_PATH_IMAGE163
Figure DEST_PATH_IMAGE164
，
Figure 812693DEST_PATH_IMAGE165
(ii) a From the above equation, an error variable is estimated
Figure DEST_PATH_IMAGE166
，
Figure 608610DEST_PATH_IMAGE167
，
Figure DEST_PATH_IMAGE168
Is bounded, i.e. stores a constant
Figure 361803DEST_PATH_IMAGE169
，
Figure DEST_PATH_IMAGE170
，
Figure 177924DEST_PATH_IMAGE171
So that
Figure DEST_PATH_IMAGE172
，
Figure 674765DEST_PATH_IMAGE173
，
Figure DEST_PATH_IMAGE174
It is true that, among other things,
Figure 590768DEST_PATH_IMAGE175
；
constructing the Lyapunov function
Figure DEST_PATH_IMAGE176
To is aligned with
Figure 514862DEST_PATH_IMAGE177
Taking the time derivative and taking into account the compensating system
Figure DEST_PATH_IMAGE178
Wherein the error variable
Figure 883526DEST_PATH_IMAGE179
(ii) a According to the Young's inequality
Figure DEST_PATH_IMAGE180
From the above formula and virtual control law
Figure 121741DEST_PATH_IMAGE181
To obtain
Figure DEST_PATH_IMAGE182
Constructing the Lyapunov function
Figure 626671DEST_PATH_IMAGE183
Is composed of
Figure DEST_PATH_IMAGE184
Wherein the error variable
Figure 721666DEST_PATH_IMAGE185
Taking into account the parameters of the mechanical arm
Figure DEST_PATH_IMAGE186
And
Figure 577627DEST_PATH_IMAGE187
is obliquely symmetrical with respect to
Figure DEST_PATH_IMAGE188
Taking the time derivative to obtain
Figure 416270DEST_PATH_IMAGE189
Wherein the error variable
Figure DEST_PATH_IMAGE190
，
Figure 41286DEST_PATH_IMAGE191
Is a time-varying function, satisfies
Figure DEST_PATH_IMAGE192
，
Figure 510445DEST_PATH_IMAGE193
Is a normal number of the blood vessel which is,
Figure DEST_PATH_IMAGE194
is a bounded function vector; according to the Young's inequality
Figure 588122DEST_PATH_IMAGE195
Wherein,
Figure DEST_PATH_IMAGE196
represent
Figure 964877DEST_PATH_IMAGE197
The identity matrix of (a); substituting the inequality into
Figure DEST_PATH_IMAGE198
To obtain
Figure 447330DEST_PATH_IMAGE199
Constructing the Lyapunov function
Figure DEST_PATH_IMAGE200
To, for
Figure 149706DEST_PATH_IMAGE201
Taking the time derivative to obtain
Figure DEST_PATH_IMAGE202
Due to the fact that
Figure 714680DEST_PATH_IMAGE203
For positive definite diagonal matrix and for instruction filter, there is
Figure DEST_PATH_IMAGE204
So that
Figure 895126DEST_PATH_IMAGE205
If true; thus by selection
Figure DEST_PATH_IMAGE206
To obtain
Figure 494734DEST_PATH_IMAGE207
According to the formula, the compound has the advantages of,
Figure DEST_PATH_IMAGE208
is asymptotically stable, i.e. when
Figure 305695DEST_PATH_IMAGE209
When the utility model is used, the water is discharged,
Figure DEST_PATH_IMAGE210
(ii) a Constructing the Lyapunov function
Figure 92386DEST_PATH_IMAGE211
To, for
Figure DEST_PATH_IMAGE212
Taking the time derivative to obtain
Figure 76522DEST_PATH_IMAGE213
Wherein,
Figure DEST_PATH_IMAGE214
Figure 530637DEST_PATH_IMAGE215
，
Figure DEST_PATH_IMAGE216
，
Figure 309238DEST_PATH_IMAGE217
Figure DEST_PATH_IMAGE218
(ii) a According to the formula, the compound has the advantages of,
Figure 848803DEST_PATH_IMAGE219
、
Figure DEST_PATH_IMAGE220
、
Figure 371052DEST_PATH_IMAGE221
is bounded; according to
Figure DEST_PATH_IMAGE222
In the knowledge that,
Figure 679673DEST_PATH_IMAGE223
is bounded and tracking error
Figure DEST_PATH_IMAGE224
Are subject to preset performance requirements, i.e.
Figure 832437DEST_PATH_IMAGE225
(ii) a Because of the desired trajectory
Figure DEST_PATH_IMAGE226
Is bounded, therefore
Figure 856369DEST_PATH_IMAGE227
And
Figure DEST_PATH_IMAGE228
is bounded, which means that
Figure 182308DEST_PATH_IMAGE229
And
Figure DEST_PATH_IMAGE230
is bounded; further obtain the
Figure 611015DEST_PATH_IMAGE231
、
Figure DEST_PATH_IMAGE232
、
Figure DEST_PATH_IMAGE233
、
Figure 200260DEST_PATH_IMAGE234
、
Figure DEST_PATH_IMAGE235
And
Figure 448839DEST_PATH_IMAGE236
is bounded and therefore the closed loop system is stable;
finally proving the existence
Figure DEST_PATH_IMAGE237
So that
Figure 312889DEST_PATH_IMAGE238
If true;
for the
Figure DEST_PATH_IMAGE239
And
Figure 799365DEST_PATH_IMAGE240
according to
Figure DEST_PATH_IMAGE241
To obtain
Figure 90670DEST_PATH_IMAGE242
From the above derivation, there are normal numbers
Figure DEST_PATH_IMAGE243
So that
Figure 826544DEST_PATH_IMAGE244
If true; because of
Figure DEST_PATH_IMAGE245
And
Figure 494286DEST_PATH_IMAGE246
so that adjacent trigger time intervals
Figure DEST_PATH_IMAGE247
(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:
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;
the established nonlinear dynamic model of the multi-joint mechanical arm with unmodeled dynamics is as follows:
Figure 632006DEST_PATH_IMAGE001
wherein,
Figure 890949DEST_PATH_IMAGE002
a vector of the angular position of the joint is represented,
Figure 848541DEST_PATH_IMAGE003
the angular velocity vector of the joint is represented,
Figure 116711DEST_PATH_IMAGE004
represents the angular acceleration vector of the joint,
Figure 108938DEST_PATH_IMAGE005
the control torque provided for the motor is,
Figure 476466DEST_PATH_IMAGE006
in order to define the inertia matrix in a symmetrical positive way,
Figure 983670DEST_PATH_IMAGE007
is a matrix of centrifugal forces and coriolis forces,
Figure 742284DEST_PATH_IMAGE008
is a gravity vector.
The state equation is:
Figure 589017DEST_PATH_IMAGE009
wherein,
Figure 455342DEST_PATH_IMAGE010
，
Figure 121947DEST_PATH_IMAGE011
，
Figure 731920DEST_PATH_IMAGE012
representing unmodeled dynamics by the expression
Figure 167580DEST_PATH_IMAGE013
。
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;
the unmodeled dynamics in the RBF neural network approximation state equation is:
Figure 204806DEST_PATH_IMAGE014
wherein,
Figure 358707DEST_PATH_IMAGE015
a vector of the excitation function is represented,
Figure 506792DEST_PATH_IMAGE016
represents an approximation error, and satisfies
Figure 62538DEST_PATH_IMAGE017
，
Figure 270665DEST_PATH_IMAGE018
Is a normal number which is a positive number,
Figure 177441DEST_PATH_IMAGE019
is an ideal weight matrix and the weight matrix is,
Figure 66900DEST_PATH_IMAGE020
is a vector of the gaussian basis function,
Figure 539470DEST_PATH_IMAGE019
and
Figure 856182DEST_PATH_IMAGE020
can be expressed as
Figure 46991DEST_PATH_IMAGE021
Wherein for
Figure 474562DEST_PATH_IMAGE022
，
Figure 67217DEST_PATH_IMAGE023
，
Figure 554830DEST_PATH_IMAGE024
The expression of the Gaussian base function is
Figure 170619DEST_PATH_IMAGE025
Wherein,
Figure 464197DEST_PATH_IMAGE026
the number of the nodes of the RBF neural network,
Figure 114622DEST_PATH_IMAGE027
is as follows
Figure 773136DEST_PATH_IMAGE028
The center vector of each of the nodes is,
Figure 672959DEST_PATH_IMAGE029
is as follows
Figure 442332DEST_PATH_IMAGE028
The gaussian base width of each node is,
Figure 9579DEST_PATH_IMAGE030
representing an exponential function.
The state observer is constructed as follows:
Figure 838995DEST_PATH_IMAGE031
wherein,
Figure 426447DEST_PATH_IMAGE032
，
Figure 61827DEST_PATH_IMAGE033
，
Figure 421265DEST_PATH_IMAGE034
，
Figure 483899DEST_PATH_IMAGE035
，
Figure 295997DEST_PATH_IMAGE036
for design parameters, and for positive numbers,
Figure 469489DEST_PATH_IMAGE037
to represent
Figure 683433DEST_PATH_IMAGE038
The estimated amount of (a) is,
Figure 916968DEST_PATH_IMAGE039
indicating angular velocity of joint
Figure 481942DEST_PATH_IMAGE040
The estimated amount of (a) is,
Figure 396808DEST_PATH_IMAGE041
and
Figure 527575DEST_PATH_IMAGE042
respectively represent
Figure 869695DEST_PATH_IMAGE043
And
Figure 718702DEST_PATH_IMAGE044
the estimated amount of (a) is,
Figure 437259DEST_PATH_IMAGE045
is the intermediate auxiliary variable.
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;
the tracking error is:
Figure 422533DEST_PATH_IMAGE051
wherein,
Figure 935554DEST_PATH_IMAGE052
for the desired joint angle position vector to be,
Figure 271857DEST_PATH_IMAGE053
in order to provide a tracking error for the mechanical arm track,
Figure 794105DEST_PATH_IMAGE054
represents a joint angle position vector;
the preset performance function is:
Figure 837148DEST_PATH_IMAGE055
wherein,
Figure 583387DEST_PATH_IMAGE056
，
Figure 344669DEST_PATH_IMAGE057
representation for limiting tracking error
Figure 201767DEST_PATH_IMAGE058
The function of the preset performance of the system,
Figure 99316DEST_PATH_IMAGE059
，
Figure 282035DEST_PATH_IMAGE060
，
Figure 265035DEST_PATH_IMAGE061
is a design parameter, and
Figure 925823DEST_PATH_IMAGE062
，
Figure 209037DEST_PATH_IMAGE060
and
Figure 234762DEST_PATH_IMAGE063
is a positive number of the bits,
Figure 767375DEST_PATH_IMAGE064
representing exponential functions, parameters
Figure 172467DEST_PATH_IMAGE061
For designing convergence time, parameters, of tracking error
Figure 106925DEST_PATH_IMAGE060
To design the convergence accuracy of the tracking error.
The unconstrained tracking error obtained by the nonlinear transformation is:
Figure 303551DEST_PATH_IMAGE046
wherein,
Figure 323459DEST_PATH_IMAGE047
，
Figure 263734DEST_PATH_IMAGE048
for the purpose of an unconstrained tracking error,
Figure 52698DEST_PATH_IMAGE049
and
Figure 951384DEST_PATH_IMAGE050
respectively, a preset performance function and an initial value of the tracking error.
If the performance function is preset
Figure 396272DEST_PATH_IMAGE066
Then, then
Figure 936975DEST_PATH_IMAGE067
Namely, the control method of the present invention degenerates to the conventional non-default performance control method.
Step 4, designing an instruction filter with a low-order compensation system according to an unconstrained tracking error and a reverse-thrust design method;
the instruction filter is:
Figure 783708DEST_PATH_IMAGE068
wherein,
Figure 384453DEST_PATH_IMAGE069
in order to design the parameters of the device,
Figure 316637DEST_PATH_IMAGE070
is a virtual control law with the expression of
Figure 864293DEST_PATH_IMAGE071
Wherein,
Figure 627850DEST_PATH_IMAGE072
representing design parameters and being a positive definite matrix, unconstrained tracking error vector
Figure 602759DEST_PATH_IMAGE073
，
Figure 818977DEST_PATH_IMAGE074
，
Figure 904745DEST_PATH_IMAGE075
To a
Figure 522808DEST_PATH_IMAGE076
，
Figure 668618DEST_PATH_IMAGE077
，
Figure 372132DEST_PATH_IMAGE078
，
Figure 996011DEST_PATH_IMAGE079
To account for the tracking error of the compensated signal, the expression is
Figure 734160DEST_PATH_IMAGE080
Wherein,
Figure 50872DEST_PATH_IMAGE081
to compensate the signal, it is generated by a low-order compensation system; constructing a low order compensation system of
Figure 444944DEST_PATH_IMAGE082
Wherein,
Figure 934832DEST_PATH_IMAGE083
the design parameters are represented by a number of parameters,
Figure 199591DEST_PATH_IMAGE084
，
Figure 749521DEST_PATH_IMAGE085
representing a symbolic function.
Step 5, constructing unmodeled dynamics of the RBF neural network approximation control loop to obtain an estimation model of the unmodeled dynamics;
the unmodeled dynamics of the control loop are:
Figure 365310DEST_PATH_IMAGE087
constructing RBF neural network to obtain unmodeled dynamic estimation model
Figure 658888DEST_PATH_IMAGE086
Wherein,
Figure 40803DEST_PATH_IMAGE088
a vector of the excitation function is represented,
Figure 699318DEST_PATH_IMAGE089
represents an approximation error, and satisfies
Figure 864720DEST_PATH_IMAGE090
，
Figure 634093DEST_PATH_IMAGE091
Is a normal number of the cells, and,
Figure 201340DEST_PATH_IMAGE092
is an ideal weight matrix and the weight matrix is,
Figure 765177DEST_PATH_IMAGE093
is a vector of the gaussian basis function,
Figure 417875DEST_PATH_IMAGE094
and
Figure 990939DEST_PATH_IMAGE093
can be expressed as
Figure 147114DEST_PATH_IMAGE095
Wherein, for
Figure 147431DEST_PATH_IMAGE096
，
Figure 225108DEST_PATH_IMAGE097
，
Figure 664180DEST_PATH_IMAGE098
The expression of the Gaussian base function is
Figure 878123DEST_PATH_IMAGE099
Wherein,
Figure 111659DEST_PATH_IMAGE100
the number of the nodes of the RBF neural network,
Figure 676632DEST_PATH_IMAGE101
is as follows
Figure 591499DEST_PATH_IMAGE102
The center vector of each of the nodes is,
Figure 722266DEST_PATH_IMAGE103
is as follows
Figure 64385DEST_PATH_IMAGE102
The gaussian-based width of an individual node,
Figure 913393DEST_PATH_IMAGE104
representing an exponential function.
If there is no instruction filter, the virtual control law is needed
Figure 366371DEST_PATH_IMAGE105
The time derivative is taken as a function of time,
Figure 617223DEST_PATH_IMAGE106
is a variable quantity
Figure 395824DEST_PATH_IMAGE107
、
Figure 669810DEST_PATH_IMAGE108
、
Figure 988796DEST_PATH_IMAGE109
、
Figure 31838DEST_PATH_IMAGE110
、
Figure 778077DEST_PATH_IMAGE111
、
Figure 273781DEST_PATH_IMAGE112
、
Figure 396458DEST_PATH_IMAGE113
、
Figure 285217DEST_PATH_IMAGE114
Is very complex. The unmodeled dynamics of the control loop is
Figure 405620DEST_PATH_IMAGE115
The 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:
Figure 716516DEST_PATH_IMAGE116
wherein,
Figure 314987DEST_PATH_IMAGE117
，
Figure 332622DEST_PATH_IMAGE118
is a positive integer and is a non-zero integer,
Figure 686243DEST_PATH_IMAGE119
、
Figure 156539DEST_PATH_IMAGE120
and
Figure 621018DEST_PATH_IMAGE121
to design parameters and satisfy
Figure 227580DEST_PATH_IMAGE122
，
Figure 752102DEST_PATH_IMAGE123
Is a function of the hyperbolic tangent,
Figure 975273DEST_PATH_IMAGE124
，
Figure 915547DEST_PATH_IMAGE125
，
Figure 704512DEST_PATH_IMAGE126
、
Figure 337618DEST_PATH_IMAGE127
、
Figure 579244DEST_PATH_IMAGE128
and
Figure 323209DEST_PATH_IMAGE129
are respectively as
Figure 232259DEST_PATH_IMAGE130
、
Figure 770688DEST_PATH_IMAGE131
、
Figure 968451DEST_PATH_IMAGE132
And
Figure 312845DEST_PATH_IMAGE133
to (1)
Figure 14084DEST_PATH_IMAGE134
A component;
law of virtual control
Figure 785731DEST_PATH_IMAGE135
Is expressed as
Figure 205211DEST_PATH_IMAGE136
Wherein,
Figure 353296DEST_PATH_IMAGE137
representing design parameters, and is a positive definite matrix,
Figure 174621DEST_PATH_IMAGE138
and
Figure 320432DEST_PATH_IMAGE139
in order to design the parameters of the device,
Figure 23946DEST_PATH_IMAGE140
，
Figure 644895DEST_PATH_IMAGE141
and
Figure 383044DEST_PATH_IMAGE142
respectively represent
Figure 434177DEST_PATH_IMAGE143
And
Figure 624987DEST_PATH_IMAGE144
is 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
Figure 52557DEST_PATH_IMAGE145
，
Figure 645212DEST_PATH_IMAGE146
，
Figure 398405DEST_PATH_IMAGE147
Wherein, in the process,
Figure 14194DEST_PATH_IMAGE148
is in a state
Figure 42193DEST_PATH_IMAGE149
(ii) an estimate of (d); for estimation error variable
Figure 427038DEST_PATH_IMAGE150
And
Figure 147869DEST_PATH_IMAGE151
respectively taking time derivatives of
Figure 250954DEST_PATH_IMAGE248
Wherein,
Figure 82644DEST_PATH_IMAGE153
，
Figure 587575DEST_PATH_IMAGE154
according to the characteristics of a Gaussian base function
Figure 416990DEST_PATH_IMAGE155
Wherein
Figure 69689DEST_PATH_IMAGE156
is a normal number, further obtained
Figure 377173DEST_PATH_IMAGE157
The following Lyapunov function is constructed
Figure 798927DEST_PATH_IMAGE158
To analyze the stability of the observer
Figure 799244DEST_PATH_IMAGE159
To lyapunov function
Figure 876922DEST_PATH_IMAGE158
Taking the time derivative to obtain
Figure 315993DEST_PATH_IMAGE160
According to the Young inequality
Figure DEST_PATH_IMAGE249
According to the inequality
Figure 795516DEST_PATH_IMAGE162
Wherein,
Figure 966735DEST_PATH_IMAGE163
Figure 328446DEST_PATH_IMAGE164
，
Figure 977733DEST_PATH_IMAGE165
(ii) a From the above equation, an error variable is estimated
Figure 374079DEST_PATH_IMAGE166
，
Figure 716199DEST_PATH_IMAGE167
，
Figure 565206DEST_PATH_IMAGE168
Is bounded, i.e. stores a constant
Figure 286693DEST_PATH_IMAGE169
，
Figure 475229DEST_PATH_IMAGE170
，
Figure 50567DEST_PATH_IMAGE171
So that
Figure 324553DEST_PATH_IMAGE172
，
Figure 643539DEST_PATH_IMAGE173
，
Figure 952161DEST_PATH_IMAGE174
It is true that, among other things,
Figure 636083DEST_PATH_IMAGE175
；
constructing Lyapunov functions
Figure 194103DEST_PATH_IMAGE176
To, for
Figure 254463DEST_PATH_IMAGE177
Taking the time derivative and taking into account the compensating system
Figure 214329DEST_PATH_IMAGE178
Wherein the error variable
Figure 69152DEST_PATH_IMAGE179
(ii) a According to the Young's inequality
Figure 380048DEST_PATH_IMAGE180
From the above formula and virtual control law
Figure 978520DEST_PATH_IMAGE181
To obtain
Figure 792892DEST_PATH_IMAGE182
Constructing the Lyapunov function
Figure 349775DEST_PATH_IMAGE183
Is composed of
Figure 820071DEST_PATH_IMAGE184
Wherein the error variable
Figure 18971DEST_PATH_IMAGE185
Taking into account the parameters of the mechanical arm
Figure 891112DEST_PATH_IMAGE186
And
Figure 415634DEST_PATH_IMAGE187
is obliquely symmetrical with respect to
Figure 107647DEST_PATH_IMAGE188
Taking the time derivative to obtain
Figure 375817DEST_PATH_IMAGE189
Wherein the error variable
Figure 102465DEST_PATH_IMAGE190
，
Figure 532309DEST_PATH_IMAGE191
Is a time-varying function, satisfies
Figure 242776DEST_PATH_IMAGE192
，
Figure 986741DEST_PATH_IMAGE193
Is a normal number, and is,
Figure 630212DEST_PATH_IMAGE194
is a bounded function vector; according to the Young's inequality
Figure 434220DEST_PATH_IMAGE195
Wherein,
Figure 163142DEST_PATH_IMAGE196
to represent
Figure 707868DEST_PATH_IMAGE197
The identity matrix of (1); substituting the inequality into
Figure 674687DEST_PATH_IMAGE198
To obtain
Figure 446334DEST_PATH_IMAGE199
Constructing the Lyapunov function
Figure 865814DEST_PATH_IMAGE200
To, for
Figure 13898DEST_PATH_IMAGE201
Taking the time derivative to obtain
Figure 569645DEST_PATH_IMAGE202
Due to the fact that
Figure 512193DEST_PATH_IMAGE203
For positive definite diagonal matrix and for instruction filter, there is
Figure 153390DEST_PATH_IMAGE204
So that
Figure 105165DEST_PATH_IMAGE205
If true; thus by selection
Figure 515418DEST_PATH_IMAGE206
To obtain
Figure 97709DEST_PATH_IMAGE207
According to the formula, the compound has the advantages of,
Figure 288519DEST_PATH_IMAGE208
is asymptotically stable, i.e. when
Figure 716089DEST_PATH_IMAGE209
When the temperature of the water is higher than the set temperature,
Figure 308745DEST_PATH_IMAGE210
(ii) a Constructing Lyapunov functions
Figure 796358DEST_PATH_IMAGE211
To, for
Figure 208885DEST_PATH_IMAGE212
Taking the time derivative to obtain
Figure 440146DEST_PATH_IMAGE213
Wherein,
Figure 887308DEST_PATH_IMAGE214
Figure 545822DEST_PATH_IMAGE215
，
Figure 914486DEST_PATH_IMAGE216
，
Figure 746176DEST_PATH_IMAGE217
Figure 985528DEST_PATH_IMAGE218
(ii) a According to the formula, the compound has the advantages of,
Figure 877260DEST_PATH_IMAGE219
、
Figure 202062DEST_PATH_IMAGE220
、
Figure 837443DEST_PATH_IMAGE221
is bounded; according to
Figure 196880DEST_PATH_IMAGE222
In the knowledge that,
Figure 259514DEST_PATH_IMAGE223
is bounded and tracking error
Figure 71612DEST_PATH_IMAGE224
Are subject to predetermined performance requirements, i.e.
Figure 245105DEST_PATH_IMAGE225
(ii) a Because of the desired trajectory
Figure 739276DEST_PATH_IMAGE226
Is bounded, therefore
Figure 910494DEST_PATH_IMAGE227
And
Figure 272206DEST_PATH_IMAGE228
is bounded, which means that
Figure 187072DEST_PATH_IMAGE229
And
Figure 583418DEST_PATH_IMAGE230
is bounded; further obtain the
Figure 925538DEST_PATH_IMAGE231
、
Figure 508966DEST_PATH_IMAGE232
、
Figure 227523DEST_PATH_IMAGE233
、
Figure 681638DEST_PATH_IMAGE234
、
Figure 256976DEST_PATH_IMAGE235
And
Figure 265384DEST_PATH_IMAGE236
is bounded and therefore the closed loop system is stable;
finally proving the existence
Figure 584369DEST_PATH_IMAGE237
So that
Figure 627412DEST_PATH_IMAGE238
If true;
for the
Figure 373651DEST_PATH_IMAGE239
And
Figure 400513DEST_PATH_IMAGE240
according to
Figure 195293DEST_PATH_IMAGE241
To obtain
Figure 420738DEST_PATH_IMAGE242
From the above derivation, there are normal numbers
Figure 275562DEST_PATH_IMAGE243
So that
Figure 524141DEST_PATH_IMAGE244
If true; because of the fact that
Figure 184929DEST_PATH_IMAGE245
And
Figure 202564DEST_PATH_IMAGE246
so that adjacent trigger time intervals
Figure 290605DEST_PATH_IMAGE247
(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 theory
Figure 760901DEST_PATH_IMAGE247
Wherein
Figure 225380DEST_PATH_IMAGE250
is 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:
Figure DEST_PATH_IMAGE251
wherein,
Figure 363101DEST_PATH_IMAGE252
，
Figure DEST_PATH_IMAGE253
and
Figure 825306DEST_PATH_IMAGE254
the angular positions of the joint 1 and the joint 2 respectively,
Figure DEST_PATH_IMAGE255
，
Figure 248810DEST_PATH_IMAGE256
and
Figure DEST_PATH_IMAGE257
the angular velocities of the joint 1 and the joint 2 respectively,
Figure 720242DEST_PATH_IMAGE258
，
Figure DEST_PATH_IMAGE259
and
Figure 712469DEST_PATH_IMAGE260
control moments, inertia matrices, of joints 1 and 2, respectively
Figure DEST_PATH_IMAGE261
Centrifugal and coriolis force matrices
Figure 345576DEST_PATH_IMAGE262
And the gravity vector
Figure DEST_PATH_IMAGE263
Respectively as follows:
Figure 56043DEST_PATH_IMAGE264
Figure DEST_PATH_IMAGE265
Figure 65587DEST_PATH_IMAGE266
wherein,
Figure DEST_PATH_IMAGE267
，
Figure 177900DEST_PATH_IMAGE268
，
Figure DEST_PATH_IMAGE269
，
Figure 247487DEST_PATH_IMAGE270
，
Figure DEST_PATH_IMAGE271
，
Figure 179671DEST_PATH_IMAGE272
. The simulation time was set to 16 seconds.
The desired trajectory is set to
Figure DEST_PATH_IMAGE273
The mechanical arm is in an initial state
Figure 992906DEST_PATH_IMAGE274
，
Figure DEST_PATH_IMAGE275
，
Figure 694146DEST_PATH_IMAGE276
。
Control law parameter set to
Figure DEST_PATH_IMAGE277
，
Figure 137896DEST_PATH_IMAGE278
，
Figure DEST_PATH_IMAGE279
，
Figure 557376DEST_PATH_IMAGE280
，
Figure DEST_PATH_IMAGE281
，
Figure 908723DEST_PATH_IMAGE282
，
Figure DEST_PATH_IMAGE283
，
Figure 998558DEST_PATH_IMAGE284
，
Figure DEST_PATH_IMAGE285
，
Figure 409948DEST_PATH_IMAGE286
，
Figure DEST_PATH_IMAGE287
，
Figure 316724DEST_PATH_IMAGE288
，
Figure DEST_PATH_IMAGE289
，
Figure 471761DEST_PATH_IMAGE290
，
Figure DEST_PATH_IMAGE291
，
Figure 147593DEST_PATH_IMAGE292
，
Figure DEST_PATH_IMAGE293
，
Figure 933147DEST_PATH_IMAGE294
，
Figure DEST_PATH_IMAGE295
，
Figure 327219DEST_PATH_IMAGE296
Initial value of state observer is set to
Figure DEST_PATH_IMAGE297
，
Figure 20369DEST_PATH_IMAGE298
，
Figure DEST_PATH_IMAGE299
。
The preset performance function of the tracking error is set by the user according to the actual situation and, in this embodiment,
Figure 816286DEST_PATH_IMAGE300
，
Figure DEST_PATH_IMAGE301
，
Figure 303899DEST_PATH_IMAGE302
。
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
Figure DEST_PATH_IMAGE303
(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 invention
Figure 185268DEST_PATH_IMAGE304
And
Figure DEST_PATH_IMAGE305
the weight norm of the neural network can be known from the graph
Figure 947687DEST_PATH_IMAGE304
And
Figure 332532DEST_PATH_IMAGE305
the 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:
Figure 53364DEST_PATH_IMAGE300
，
Figure 890870DEST_PATH_IMAGE301
，
Figure 722559DEST_PATH_IMAGE302
. Second parameter setting case:
Figure 227490DEST_PATH_IMAGE306
，
Figure DEST_PATH_IMAGE307
，
Figure 585135DEST_PATH_IMAGE308
. 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 is
Figure DEST_PATH_IMAGE309
The expression of the integral absolute error is
Figure 175516DEST_PATH_IMAGE310
Wherein
Figure DEST_PATH_IMAGE311
，
Figure 14159DEST_PATH_IMAGE312
as a matter of time, the time is,
Figure DEST_PATH_IMAGE313
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
Figure 639175DEST_PATH_IMAGE314
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:
Figure 959118DEST_PATH_IMAGE001
wherein,
Figure 345100DEST_PATH_IMAGE002
a vector of the angular position of the joint is represented,
Figure 390416DEST_PATH_IMAGE003
the angular velocity vector of the joint is represented,
Figure 113522DEST_PATH_IMAGE004
represents the angular acceleration vector of the joint,
Figure 396736DEST_PATH_IMAGE005
the control torque provided for the motor is,
Figure 484777DEST_PATH_IMAGE006
in order to define the inertia matrix in a symmetrical positive way,
Figure 328974DEST_PATH_IMAGE007
is a matrix of centrifugal forces and coriolis forces,
Figure 996716DEST_PATH_IMAGE008
is a gravity vector.
3. The state observer-based multi-joint mechanical arm event-driven control method according to claim 1, characterized in that: the state equation in step 1 is:
Figure 931174DEST_PATH_IMAGE009
wherein,
Figure 518013DEST_PATH_IMAGE010
，
Figure 475605DEST_PATH_IMAGE011
，
Figure 478196DEST_PATH_IMAGE012
representing unmodeled dynamics by the expression
Figure 345789DEST_PATH_IMAGE013
。
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:
Figure 978896DEST_PATH_IMAGE014
wherein,
Figure 486100DEST_PATH_IMAGE015
a vector of the excitation function is represented,
Figure 354699DEST_PATH_IMAGE016
represents an approximation error, and satisfies
Figure 201432DEST_PATH_IMAGE017
，
Figure 802178DEST_PATH_IMAGE018
Is a normal number which is a positive number,
Figure 373843DEST_PATH_IMAGE019
is an ideal weight matrix of the weight values,
Figure 655919DEST_PATH_IMAGE020
is a vector of the gaussian basis function,
Figure 747372DEST_PATH_IMAGE019
and
Figure 784598DEST_PATH_IMAGE020
can be expressed as
Figure 938499DEST_PATH_IMAGE021
Wherein, for
Figure 165212DEST_PATH_IMAGE022
，
Figure 783275DEST_PATH_IMAGE023
，
Figure 929086DEST_PATH_IMAGE024
The expression of the Gaussian base function is
Figure 960496DEST_PATH_IMAGE025
Wherein,
Figure 646692DEST_PATH_IMAGE026
the number of the nodes of the RBF neural network,
Figure 322524DEST_PATH_IMAGE027
is as follows
Figure 278716DEST_PATH_IMAGE028
The center vector of each of the nodes is,
Figure 469526DEST_PATH_IMAGE029
is as follows
Figure 162676DEST_PATH_IMAGE028
The gaussian-based width of an individual node,
Figure 552069DEST_PATH_IMAGE030
representing an exponential function.
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:
Figure 101999DEST_PATH_IMAGE031
wherein,
Figure 983367DEST_PATH_IMAGE032
，
Figure 89995DEST_PATH_IMAGE033
，
Figure 537157DEST_PATH_IMAGE034
，
Figure 195671DEST_PATH_IMAGE035
，
Figure 423390DEST_PATH_IMAGE036
for design parameters, and for positive numbers,
Figure 255080DEST_PATH_IMAGE037
to represent
Figure 25590DEST_PATH_IMAGE038
The estimated amount of (a) is,
Figure 963328DEST_PATH_IMAGE039
indicating angular velocity of joint
Figure 616026DEST_PATH_IMAGE040
The estimated amount of (a) is,
Figure 454669DEST_PATH_IMAGE041
and
Figure 673160DEST_PATH_IMAGE042
respectively represent
Figure 939057DEST_PATH_IMAGE043
And
Figure 813472DEST_PATH_IMAGE044
is measured in a time-domain manner,
Figure 65593DEST_PATH_IMAGE045
is the intermediate auxiliary variable.
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
Figure 545116DEST_PATH_IMAGE046
Wherein,
Figure 778651DEST_PATH_IMAGE047
，
Figure 202679DEST_PATH_IMAGE048
for the purpose of an unconstrained tracking error,
Figure 179862DEST_PATH_IMAGE049
and
Figure 513892DEST_PATH_IMAGE050
respectively setting initial values of a preset performance function and a tracking error;
wherein, the tracking error is:
Figure 253350DEST_PATH_IMAGE051
wherein,
Figure 305620DEST_PATH_IMAGE052
for the desired joint angle position vector to be,
Figure 820915DEST_PATH_IMAGE053
in order to provide a tracking error for the mechanical arm track,
Figure 134084DEST_PATH_IMAGE054
represents a joint angle position vector;
the preset performance function is:
Figure 709422DEST_PATH_IMAGE055
wherein,
Figure 248988DEST_PATH_IMAGE056
，
Figure 381023DEST_PATH_IMAGE057
representation for limiting tracking error
Figure 486382DEST_PATH_IMAGE058
The function of the preset performance of the system,
Figure 435884DEST_PATH_IMAGE059
，
Figure 56221DEST_PATH_IMAGE060
，
Figure 178898DEST_PATH_IMAGE061
is a design parameter, and
Figure 342026DEST_PATH_IMAGE062
，
Figure 570751DEST_PATH_IMAGE060
and
Figure 84909DEST_PATH_IMAGE063
is a positive number, and the number of the positive number,
Figure 745697DEST_PATH_IMAGE064
representing an exponential function.
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:
Figure 622386DEST_PATH_IMAGE065
wherein,
Figure 179270DEST_PATH_IMAGE066
in order to design the parameters of the device,
Figure 711882DEST_PATH_IMAGE067
is a virtual control law, and the expression is
Figure 989411DEST_PATH_IMAGE068
Wherein,
Figure 861552DEST_PATH_IMAGE069
representing design parameters and being a positive definite matrix, unconstrained tracking error vector
Figure 386074DEST_PATH_IMAGE070
，
Figure 733879DEST_PATH_IMAGE071
，
Figure 674153DEST_PATH_IMAGE072
To a
Figure 463118DEST_PATH_IMAGE073
，
Figure 735705DEST_PATH_IMAGE074
，
Figure 915013DEST_PATH_IMAGE075
，
Figure 49191DEST_PATH_IMAGE076
To account for the tracking error of the compensated signal, the expression is
Figure 958242DEST_PATH_IMAGE077
Wherein,
Figure 762250DEST_PATH_IMAGE078
to compensate the signal, it is generated by a low-order compensation system; constructing a low order compensation system of
Figure 569800DEST_PATH_IMAGE079
Wherein,
Figure 914193DEST_PATH_IMAGE080
the design parameters are represented by a number of parameters,
Figure 881012DEST_PATH_IMAGE081
，
Figure 714976DEST_PATH_IMAGE082
presentation symbolA function.
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
Figure 400035DEST_PATH_IMAGE083
Wherein,
Figure 548120DEST_PATH_IMAGE084
for the unmodeled dynamics of the control loop,
Figure 477768DEST_PATH_IMAGE085
a vector of the excitation function is represented,
Figure 889157DEST_PATH_IMAGE086
represents an approximation error, and satisfies
Figure 592671DEST_PATH_IMAGE087
，
Figure 341184DEST_PATH_IMAGE088
Is a normal number which is a positive number,
Figure 282596DEST_PATH_IMAGE089
is an ideal weight matrix and the weight matrix is,
Figure 396045DEST_PATH_IMAGE090
is a vector of the gaussian basis function,
Figure 665484DEST_PATH_IMAGE091
and
Figure 93054DEST_PATH_IMAGE090
can be expressed as
Figure 685709DEST_PATH_IMAGE092
Wherein, for
Figure 563535DEST_PATH_IMAGE093
，
Figure 179324DEST_PATH_IMAGE094
，
Figure 207323DEST_PATH_IMAGE095
The expression of the Gaussian base function is
Figure 231649DEST_PATH_IMAGE096
Wherein,
Figure 890163DEST_PATH_IMAGE097
the number of the nodes of the RBF neural network,
Figure 383462DEST_PATH_IMAGE098
is as follows
Figure 215151DEST_PATH_IMAGE099
The center vector of each of the nodes is,
Figure 454503DEST_PATH_IMAGE100
is as follows
Figure 424864DEST_PATH_IMAGE099
The gaussian-based width of an individual node,
Figure 77562DEST_PATH_IMAGE101
representing an exponential function.
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:
Figure 385047DEST_PATH_IMAGE102
wherein,
Figure 134697DEST_PATH_IMAGE103
，
Figure 197331DEST_PATH_IMAGE104
is a positive integer and is a non-zero integer,
Figure 9429DEST_PATH_IMAGE105
、
Figure 25664DEST_PATH_IMAGE106
and
Figure 301925DEST_PATH_IMAGE107
to design parameters and satisfy
Figure 473143DEST_PATH_IMAGE108
，
Figure 162751DEST_PATH_IMAGE109
Is a function of the hyperbolic tangent,
Figure 874355DEST_PATH_IMAGE110
，
Figure 473963DEST_PATH_IMAGE111
，
Figure 691449DEST_PATH_IMAGE112
、
Figure 540456DEST_PATH_IMAGE113
、
Figure 259014DEST_PATH_IMAGE114
and
Figure 572183DEST_PATH_IMAGE115
are respectively as
Figure 147521DEST_PATH_IMAGE116
、
Figure 687087DEST_PATH_IMAGE117
、
Figure 317657DEST_PATH_IMAGE118
And
Figure 626279DEST_PATH_IMAGE119
to (1) a
Figure 372518DEST_PATH_IMAGE120
A component;
law of virtual control
Figure 992855DEST_PATH_IMAGE121
Is expressed as
Figure 318794DEST_PATH_IMAGE122
Wherein,
Figure 278660DEST_PATH_IMAGE123
representing design parameters, and is a positive definite matrix,
Figure 8850DEST_PATH_IMAGE124
and
Figure 523008DEST_PATH_IMAGE125
in order to design the parameters of the device,
Figure 183796DEST_PATH_IMAGE126
，
Figure 326065DEST_PATH_IMAGE127
and
Figure 617369DEST_PATH_IMAGE128
respectively represent
Figure 149981DEST_PATH_IMAGE129
And
Figure 926045DEST_PATH_IMAGE130
is measured.
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
Figure 798186DEST_PATH_IMAGE131
，
Figure 322708DEST_PATH_IMAGE132
，
Figure 404934DEST_PATH_IMAGE133
Wherein
Figure 610787DEST_PATH_IMAGE134
is in a state
Figure 478380DEST_PATH_IMAGE135
(ii) an estimate of (d); for the estimated error variable
Figure 908224DEST_PATH_IMAGE136
And
Figure 353112DEST_PATH_IMAGE137
respectively taking time derivatives of
Figure 159394DEST_PATH_IMAGE138
Wherein,
Figure 130761DEST_PATH_IMAGE139
，
Figure 934769DEST_PATH_IMAGE140
according to the characteristics of a Gaussian base function
Figure 663691DEST_PATH_IMAGE141
Wherein
Figure 850827DEST_PATH_IMAGE142
is a normal number, further obtained
Figure 552067DEST_PATH_IMAGE143
The following Lyapunov function is constructed
Figure 651610DEST_PATH_IMAGE144
To analyze the stability of the observer
Figure 133407DEST_PATH_IMAGE145
To lyapunov function
Figure 219175DEST_PATH_IMAGE144
Taking the time derivative to obtain
Figure 915867DEST_PATH_IMAGE146
According to the Young inequality
Figure 858415DEST_PATH_IMAGE147
According to the above inequality
Figure 499612DEST_PATH_IMAGE148
Wherein,
Figure 779283DEST_PATH_IMAGE149
Figure 251853DEST_PATH_IMAGE150
，
Figure 568565DEST_PATH_IMAGE151
(ii) a From the above equation, an error variable is estimated
Figure 336538DEST_PATH_IMAGE152
，
Figure 826426DEST_PATH_IMAGE153
，
Figure 356764DEST_PATH_IMAGE154
Is bounded, i.e. stores a normal number
Figure 234590DEST_PATH_IMAGE155
，
Figure 647117DEST_PATH_IMAGE156
，
Figure 143957DEST_PATH_IMAGE157
So that
Figure 404169DEST_PATH_IMAGE158
，
Figure 125000DEST_PATH_IMAGE159
，
Figure 493664DEST_PATH_IMAGE160
It is true that, among other things,
Figure 387671DEST_PATH_IMAGE161
；
constructing Lyapunov functions
Figure 689339DEST_PATH_IMAGE162
To, for
Figure 518755DEST_PATH_IMAGE163
Taking the time derivative and taking into account the compensating system
Figure 483038DEST_PATH_IMAGE164
Wherein the error variable
Figure 118418DEST_PATH_IMAGE165
(ii) a According to the Young's inequality
Figure 743435DEST_PATH_IMAGE166
From the above formula and virtual control law
Figure 868386DEST_PATH_IMAGE167
To obtain
Figure 742801DEST_PATH_IMAGE168
Constructing the Lyapunov function
Figure 853976DEST_PATH_IMAGE169
Is composed of
Figure 208865DEST_PATH_IMAGE170
Wherein the error variable
Figure 442401DEST_PATH_IMAGE171
Taking into account the parameters of the mechanical arm
Figure 741795DEST_PATH_IMAGE172
And
Figure 46874DEST_PATH_IMAGE173
oblique symmetry of (2), to
Figure 646483DEST_PATH_IMAGE174
Taking the time derivative to obtain
Figure 50919DEST_PATH_IMAGE175
Wherein the error variable
Figure 211511DEST_PATH_IMAGE176
，
Figure 930068DEST_PATH_IMAGE177
Is a time-varying function, satisfies
Figure 508817DEST_PATH_IMAGE178
，
Figure 84155DEST_PATH_IMAGE179
Is a normal number, and is,
Figure 92562DEST_PATH_IMAGE180
is a bounded function vector; according to the Young's inequality
Figure 490177DEST_PATH_IMAGE181
Wherein,
Figure 595536DEST_PATH_IMAGE182
to represent
Figure 279458DEST_PATH_IMAGE183
The identity matrix of (1); substituting the inequality into
Figure 430954DEST_PATH_IMAGE184
To obtain
Figure 288052DEST_PATH_IMAGE185
Constructing the Lyapunov function
Figure 451180DEST_PATH_IMAGE186
To, for
Figure 945484DEST_PATH_IMAGE187
Taking the time derivative to obtain
Figure 990800DEST_PATH_IMAGE188
Due to the fact that
Figure 854851DEST_PATH_IMAGE189
For positive definite diagonal matrixAnd for the instruction filter, there is
Figure 997119DEST_PATH_IMAGE190
So that
Figure 85161DEST_PATH_IMAGE191
If true; thus by selection
Figure 821036DEST_PATH_IMAGE192
To obtain
Figure 98565DEST_PATH_IMAGE193
According to the formula, the compound has the advantages of,
Figure 33023DEST_PATH_IMAGE194
is asymptotically stable, i.e. when
Figure 495228DEST_PATH_IMAGE195
When the temperature of the water is higher than the set temperature,
Figure 843033DEST_PATH_IMAGE196
(ii) a Constructing the Lyapunov function
Figure 48886DEST_PATH_IMAGE197
To, for
Figure 837851DEST_PATH_IMAGE198
Taking the time derivative to obtain
Figure 844859DEST_PATH_IMAGE199
Wherein,
Figure 289746DEST_PATH_IMAGE200
Figure 96028DEST_PATH_IMAGE201
，
Figure 67395DEST_PATH_IMAGE202
，
Figure 871403DEST_PATH_IMAGE203
Figure 600325DEST_PATH_IMAGE204
(ii) a According to the formula, the compound has the advantages of,
Figure 288926DEST_PATH_IMAGE205
、
Figure 724587DEST_PATH_IMAGE206
、
Figure 761813DEST_PATH_IMAGE207
is bounded; according to
Figure 305927DEST_PATH_IMAGE208
In the knowledge that,
Figure 391695DEST_PATH_IMAGE209
is bounded and tracking error
Figure 9758DEST_PATH_IMAGE210
Are subject to preset performance requirements, i.e.
Figure 795049DEST_PATH_IMAGE211
(ii) a Because of the desired trajectory
Figure 436246DEST_PATH_IMAGE212
Is bounded, therefore
Figure 715917DEST_PATH_IMAGE213
And
Figure 188487DEST_PATH_IMAGE214
is bounded, which means that
Figure 505199DEST_PATH_IMAGE215
And
Figure 774637DEST_PATH_IMAGE216
is bounded; further obtain the
Figure 326841DEST_PATH_IMAGE217
、
Figure 857180DEST_PATH_IMAGE218
、
Figure 141531DEST_PATH_IMAGE219
、
Figure 396800DEST_PATH_IMAGE220
、
Figure 628062DEST_PATH_IMAGE221
And
Figure 75223DEST_PATH_IMAGE222
is bounded and therefore the closed loop system is stable;
finally proving the existence
Figure 123951DEST_PATH_IMAGE223
So that
Figure 961457DEST_PATH_IMAGE224
Establishing;
for the
Figure 871775DEST_PATH_IMAGE225
And
Figure 439023DEST_PATH_IMAGE226
according to
Figure 268438DEST_PATH_IMAGE227
To obtain
Figure 983454DEST_PATH_IMAGE228
From the above derivation, there are normal numbers
Figure 618834DEST_PATH_IMAGE229
So that
Figure 978271DEST_PATH_IMAGE230
If true; because of the fact that
Figure 641507DEST_PATH_IMAGE231
And
Figure 515922DEST_PATH_IMAGE232
so that adjacent trigger time intervals
Figure 627097DEST_PATH_IMAGE233
(ii) a Therefore, the event-driven control law designed by the invention is reasonable, namely, the Seno behavior is avoided.
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