CN113534666B - Trajectory tracking control method of single-joint mechanical arm system under multi-target constraint - Google Patents

Trajectory tracking control method of single-joint mechanical arm system under multi-target constraint Download PDF

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CN113534666B
CN113534666B CN202110866514.5A CN202110866514A CN113534666B CN 113534666 B CN113534666 B CN 113534666B CN 202110866514 A CN202110866514 A CN 202110866514A CN 113534666 B CN113534666 B CN 113534666B
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joint mechanical
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宋晓娜
孙鹏
宋帅
李星儒
张其源
孙祥亮
胡东肖
张震
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Henan University of Science and Technology
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Abstract

The invention provides a track tracking control method of a single-joint mechanical arm system under preset performance, which is based on the current situation that few researches are based on a fuzzy state observer, a fixed time command filter triggered by a self-adaptive event and a barrier Lyapunov function method are combined and applied to a nonlinear system of a single-joint mechanical arm, the track tracking control is researched by taking the nonlinear system with typical repeated motion, such as the single-joint mechanical arm, as an object, and compared with a finite time algorithm, the track tracking control method has higher convergence speed; compared with the general self-adaptive backstepping control, the method can reduce the communication burden and the calculation amount, so that the method has higher engineering practical value for the research of the single-joint mechanical arm.

Description

Trajectory tracking control method of single-joint mechanical arm system under multi-target constraint
Technical Field
The invention relates to a control method of a single-joint mechanical arm system, in particular to a track tracking control method of the single-joint mechanical arm system under multi-target constraint.
Background
The mechanical arm is an important component of the joint robot, plays an important role in the fields of industry, manufacturing industry, national defense and military and the like, can be used for production operation in various environments with high cost, danger and complexity of alternative manpower, and gradually goes to practicality in various fields through years of research and development, for example: (1) In the civil field, for example, the etiquette robot provides welcome service, navigation information service, talent performance and the like for the public; (2) In the industrial field, such as mechanical arms for welding and reinforcing screws on an automobile production line, a rapid brick-moving and building robot on a construction site, a conveying and packaging conveying in a warehouse, an assembling robot and the like; (3) In special fields, such as explosive disposal and dangerous work for national defense and military, armed police, troops and the like; (4) The aerospace field, such as replacing human beings to clamp and mount objects in an outer space workstation. With the wide application of multi-joint mechanical arms in robots, in order to achieve comprehensive optimization of performance indexes of multi-joint mechanical arms (controlled systems), an optimal control method of the multi-joint mechanical arms gradually becomes a key point of design of the joint robots.
The self-adaptive backstepping control method is an effective algorithm capable of processing the control problem of a nonlinear system, and is mainly applied to the tracking control problem of the system. The backstepping method is actually a design method which recurs from front to back, wherein the introduced virtual control is essentially a static compensation idea, and the front subsystem must achieve the purpose of stabilization through the virtual control of the back subsystem. In practical systems, there are mostly unknown functions, and the unknown terms can be approximated by using a fuzzy logic system or a neural network. Meanwhile, in a backstepping frame, because the problem of complexity explosion of calculated amount is generated by repeated derivation of a virtual control signal, the problem is perfectly solved by introducing a dynamic surface control technology, G.Sun et al combines an adaptive fuzzy technology with DSC to eliminate the influence of uncertain nonlinearity in a system, however, the method does not consider the influence of first-order filtering error; farrell et al further propose a command filtering technique to reduce the influence of filtering errors by constructing an error compensation mechanism, but the above back-step controller based on command filtering can only achieve asymptotic stability.
Unlike asymptotic control methods, finite time control methods can ensure that tracking errors converge to a balance point quickly, and recently, y. -x.li et al have studied the finite time command filtering backstepping situation of uncertain nonlinear systems, in which the set convergence time is closely related to the initial state, but once the initial state is far from the balance point, the convergence time may be ineffective. At present, M.Chen et al firstly researches an adaptive actual fixed time tracking algorithm of a strict feedback nonlinear system, the predicted convergence time of the algorithm is irrelevant to an initial value, and a natural problem is that: how to extend these conventional non-linear controls to account for the communication burden. By introducing an event trigger control strategy, communication burden can be effectively relieved, waste of unnecessary communication resources is reduced, and W.Yang et al further solve the problem of fixed time control based on event trigger, but still cannot ignore the influence of constraint conditions in an actual system.
Related constraint problems typically occur in engineering instances such as cranes, articulated robots, etc. If these constraint problems are not properly addressed, it may degrade the performance of the system. However, the problem of multi-objective constraints has not led to much research. For example: in the material conveying process, the shortest distance and the lowest transportation cost proposed by J.Liu et al belong to multi-target constraints, so that a basic control method becomes more interesting and challenging, L.Liu et al further provides self-adaptive finite time control of a nonlinear system with multi-target constraints, and finally the stability of the system is ensured by utilizing a Lyapunov stability theory, so that the track tracking control of the single-joint mechanical arm is realized.
In summary, few studies are currently conducted on the basis of a fuzzy state observer, and a fixed-time command filter triggered by an adaptive event and a barrier Lyapunov function method are combined and applied to a nonlinear system of a single-joint mechanical arm.
Disclosure of Invention
In view of this, the present invention provides a self-adaptive fixed time command filtering tracking control strategy combining a state observer and a barrier lyapunov function for a non-strict feedback nonlinear system with multi-target constraint and an unmeasured state, and specifically provides a trajectory tracking control method for a single-joint mechanical arm system under multi-target constraint.
In order to achieve the purpose, the technical scheme adopted by the invention is as follows: the track tracking control method of the single-joint mechanical arm system under multi-target constraint comprises the following steps:
step 1, establishing a state space model of a single-joint mechanical arm according to a mathematical model of a single-joint mechanical arm system, constructing a corresponding state observer to estimate an unmeasured state, and finally performing luggage Jaconov stability analysis by referring to an observation error system;
step 2, according to the state space model of the single-joint mechanical arm system established in the step 1, introducing a barrier function to solve the multi-target constraint problem, constructing a first Lyapunov function, and setting a corresponding virtual control law and a corresponding parameter self-adaptive law;
step 3, constructing a second Lyapunov function according to the state space model of the single-joint mechanical arm system established in the step 1, and setting a corresponding virtual control law and a corresponding parameter self-adaptive law;
step 4, constructing a third Lyapunov function according to the state space model of the single-joint mechanical arm system established in the step 1, and setting a corresponding virtual control law and a corresponding parameter self-adaptive law;
and 5, introducing an event trigger strategy to reduce the communication burden on the basis of the steps, so that the single-joint mechanical arm system meets the actual fixed time stability condition, and the track tracking control of the single-joint mechanical arm system is completed.
Further, step 1 specifically includes:
step 1.1, firstly, according to a system structure diagram of the single-joint mechanical arm, establishing a nonlinear mathematical model of the single-joint mechanical arm as follows:
Figure BDA0003187648080000041
Figure BDA0003187648080000042
wherein
Figure BDA0003187648080000043
q represents the acceleration, velocity and position of the stick, respectively, v represents the torque induced by the power subsystem, u represents the control input, D =1.5kg m 2 Represents mechanical inertia, B =1Nms/rad is represented inA viscous friction coefficient at the joint, H =1 Ω represents armature resistance, M = H represents armature inductance, and L =0.2Nm/a represents a back electromotive force coefficient;
step 1.2, define the system state variable x 1 = q, system status
Figure BDA0003187648080000045
x 3 And = ν, and the output signal y of the single-joint mechanical arm control system = q, the nonlinear model of the single-joint mechanical arm system can be represented as follows:
Figure BDA0003187648080000044
wherein f is 1 (x)＝0，g 1 (x 1 )＝1，f 2 (x)＝-10sin(x 1 )-x 2 ，g 2 (x 2 )＝1，f 3 (x)＝-0.2x 2 -x 3 ，g 3 (x 3 )＝1；f 1 (x)，f 2 (x)，f 3 (x)，g 1 (x 1 )，g 2 (x 2 ) And g 3 (x 3 ) Are all in the domain of definition
Figure BDA0003187648080000051
A non-linear function with fully smooth inner surface and satisfying g 1 (x 1 )≠0，g 2 (x 2 ) Not equal to 0 and g 3 (x 3 )≠0；
Step 1.3, representing the nonlinear model of the single-joint mechanical arm system as a state space model as follows:
Figure BDA0003187648080000052
in the formula (I), the compound is shown in the specification,
Figure BDA0003187648080000053
K＝(k 1 ,k 2 ,k 3 ) T ,B i ＝(0,1,0) T ,B＝(0,0,1) T ，C＝(1,00); a is a strict Hurwitz matrix, and by choosing the appropriate K, there is a positive definite matrix Q = Q T ＞0，P＝P T Is greater than 0 and satisfies A T P+PA＝-Q；
Step 1.4, setting the corresponding state observer as follows:
Figure BDA0003187648080000054
in the formula
Figure BDA0003187648080000055
Figure BDA0003187648080000056
Each represents x = (x) 1 ,x 2 ,x 3 ) T ，f i (x) An estimated value of (d);
based on the fuzzy logic rules, we can get:
Figure BDA0003187648080000057
Figure BDA0003187648080000058
in the formula of i Which represents the minimum approximation error, is,
Figure BDA0003187648080000059
represents the optimal weight vector, if any
Figure BDA00031876480800000510
Satisfy the requirement of
Figure BDA00031876480800000511
The observation error can be expressed as
Figure BDA00031876480800000512
Wherein δ = (δ) 1 ,δ 2 ,δ 3 ) T ，
Figure BDA00031876480800000513
Step 1.5, constructing a corresponding Lyapunov function as:
Figure BDA0003187648080000061
derivation of this can yield:
Figure BDA0003187648080000062
in view of the Young's inequality and fuzzy basis functions
Figure BDA0003187648080000063
The following can be obtained:
Figure BDA0003187648080000064
wherein
Figure BDA0003187648080000065
Bringing the inequality of the above into
Figure BDA0003187648080000066
The following can be obtained:
Figure BDA0003187648080000067
in the formula (I), the compound is shown in the specification,
Figure BDA0003187648080000068
further, step 2 specifically includes:
step 2.1, the barrier function is designed as follows:
Figure BDA0003187648080000069
wherein the content of the first and second substances,
Figure BDA00031876480800000610
m i (i = 1.... N) represents a weighting coefficient;
step 2.2, defining the following coordinate transformation:
z 1 (t)＝ξ-y d ,
Figure BDA00031876480800000611
Figure BDA00031876480800000612
Figure BDA00031876480800000613
Figure BDA00031876480800000614
where xi is the barrier function, z i As systematic state error, y d As a reference signal, the reference signal is,
Figure BDA0003187648080000071
to compensate for error signals, eta i An error compensation signal;
step 2.3, the following error compensation mechanism is introduced to solve the filtering error
Figure BDA0003187648080000072
The influence of (a):
Figure BDA0003187648080000073
wherein
Figure BDA0003187648080000074
Is the output signal of a first-order filter, alpha i Input signal, beta, representing a first order filter i > 0 is a time constant; eta i (0)＝0,χ j,1 ＝1(j＝2,...,n)，k i1 ＞0,k i2 > 0 is a design parameter;
Figure BDA0003187648080000075
error compensation signal incorporating the above
Figure BDA0003187648080000076
The following can be obtained:
Figure BDA0003187648080000077
constructing the Lyapunov function
Figure BDA0003187648080000078
Wherein
Figure BDA0003187648080000079
In order to estimate the error for the parameter,
Figure BDA00031876480800000710
at the same time to V 1 The derivation can be:
Figure BDA00031876480800000711
using fuzzy basis functions
Figure BDA00031876480800000712
And can be obtained by processing the Young inequality:
Figure BDA0003187648080000081
Figure BDA0003187648080000082
where τ > 0, the above formula can be substituted:
Figure BDA0003187648080000083
law of virtual control
Figure BDA0003187648080000084
And law of parameter adaptation
Figure BDA0003187648080000085
And
Figure BDA0003187648080000086
wherein k is 11 ,k 12 ,τ,σ 1 ,c 1 ,r 1 ,
Figure BDA0003187648080000087
Are all normal numbers;
the virtual control law and the parameter adaptive law are brought into the following conditions:
Figure BDA0003187648080000088
wherein
Figure BDA0003187648080000089
Further, step 3 specifically includes:
combining the state space model of the single-joint mechanical arm system in the step 2 and the coordinate transformation, the following can be obtained:
Figure BDA00031876480800000810
wherein
Figure BDA00031876480800000811
And introducing an error compensation signal to solve the influence of filtering errors:
Figure BDA0003187648080000091
constructing a second Lyapunov function for ensuring the stability of the single-joint mechanical arm system:
Figure BDA0003187648080000092
the derivation of which is:
Figure BDA0003187648080000093
using fuzzy basis functions
Figure BDA0003187648080000094
And young inequality treatment can obtain:
Figure BDA0003187648080000095
Figure BDA0003187648080000096
replacing the corresponding formula can be:
Figure BDA0003187648080000097
law of virtual control
Figure BDA0003187648080000098
And law of parameter adaptation
Figure BDA0003187648080000099
And
Figure BDA00031876480800000910
wherein k is 21 ,k 22 ,τ,σ 2 ,c 2 ,r 2 ,
Figure BDA00031876480800000911
Are all normal numbers;
the virtual control law and the parameter adaptive law are brought into being available:
Figure BDA0003187648080000101
wherein
Figure BDA0003187648080000102
Further, step 4 specifically includes:
the state space model and the coordinate transformation of the single-joint mechanical arm system combined with the steps can obtain:
Figure BDA0003187648080000103
and introducing an error compensation signal to solve the influence of filtering errors:
Figure BDA0003187648080000104
constructing a third Lyapunov function for ensuring the stability of the single-joint mechanical arm system:
Figure BDA0003187648080000105
the derivation of which is:
Figure BDA0003187648080000106
using fuzzy basis functions as above
Figure BDA0003187648080000107
And the young inequality can be given by:
Figure BDA0003187648080000108
before setting the actual event-triggered controller u, the virtual control law α is set as follows 4 And law of parameter adaptation
Figure BDA0003187648080000109
Figure BDA00031876480800001010
Figure BDA00031876480800001011
Further, step 5 specifically includes:
defining the event trigger error as P (t) = v (t) -u (t)
Wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0003187648080000111
ρ,μ 1 ,μ 2 are all normal numbers and satisfy
Figure BDA0003187648080000112
t k ,k∈z + Representing an input update time;
at interval time t k ,t k+1 In the method, | v (t) -u (t) | < τ | u (t) | + μ is obtained based on the event-triggered control strategy 2 The controller u is set as
Figure BDA0003187648080000113
Wherein
Figure BDA0003187648080000114
The following can be obtained:
Figure BDA0003187648080000115
since 0 < 1+l 1 (t) τ < 1+ τ and
Figure BDA0003187648080000116
can obtain
Figure BDA0003187648080000117
Bringing into availability:
Figure BDA0003187648080000118
compared with the prior art, the invention has the beneficial effects that: the invention takes a nonlinear system with typical repeated motion, such as a single-joint mechanical arm, as an object to carry out the research of track tracking control, and has faster convergence speed compared with a finite time algorithm; compared with the general self-adaptive backstepping control, the method can reduce the communication burden and the calculation amount, so that the method has higher engineering practical value for the research of the single-joint mechanical arm.
Drawings
FIG. 1 is a diagram of the structure of a single joint robotic arm system and its force analysis;
FIG. 2 is a flow chart diagram of a trajectory tracking control method of a single-joint mechanical arm system under multi-target constraint according to the invention;
FIG. 3 is a trace of the output signal, the observation signal and the reference signal of the single joint mechanical arm;
FIG. 4 is a schematic view of a tracking error for a single joint robotic arm.
Detailed Description
In order to make the objects, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be described clearly and completely with reference to the accompanying drawings in the present invention, and it is obvious that the described embodiments are some embodiments of the present invention, but not all embodiments, and all other embodiments obtained by a person of ordinary skill in the art based on the embodiments of the present invention without creative efforts belong to the protection scope of the present invention.
A track tracking control method of a single-joint mechanical arm system under multi-target constraint is shown in figure 2 and comprises the following steps:
step 1, establishing a state space model of a single-joint mechanical arm according to a mathematical model of a single-joint mechanical arm system, constructing a corresponding state observer to estimate an unmeasured state, and finally performing luggage Jaconov stability analysis by referring to an observation error system;
step 2, according to the state space model of the single-joint mechanical arm system established in the step 1, introducing a barrier function to solve the multi-target constraint problem, constructing a first Lyapunov function, and setting a corresponding virtual control law and a corresponding parameter self-adaptive law;
step 3, constructing a second Lyapunov function according to the state space model of the single-joint mechanical arm system established in the step 1, and setting a corresponding virtual control law and a corresponding parameter self-adaptive law;
step 4, constructing a third Lyapunov function according to the state space model of the single-joint mechanical arm system established in the step 1, and setting a corresponding virtual control law and a corresponding parameter self-adaptive law;
and 5, introducing an event trigger strategy to reduce the communication burden on the basis of the steps, so that the system meets the actual fixed time stability condition, namely completing the track tracking control of the single-joint mechanical arm system.
The technical scheme of each step is explained in detail as follows:
step 1, establishing a state space model of the single-joint mechanical arm according to a mathematical model of a single-joint mechanical arm system, constructing a corresponding state observer to estimate an unmeasured state, and finally performing luggage Jaconov stability analysis by referring to an observation error system, wherein the method specifically comprises the following steps:
step 1.1, firstly, according to a structure diagram of a single-joint mechanical arm system, as shown in fig. 1, establishing a nonlinear mathematical model of the single-joint mechanical arm as follows:
Figure BDA0003187648080000131
Figure BDA0003187648080000132
wherein
Figure BDA0003187648080000133
q represents the acceleration, velocity and position of the stick, v represents the torque induced by the power subsystem, u represents the control input, D =1.5kg m 2 Representing mechanical inertia, B =1Nms/rad representing the viscous friction coefficient at the joint, H =1 Ω representing the armature resistance, M = H representing the armature inductance, L =0.2Nm/a representing the back emf coefficient;
step 1.2, define the system state variable x 1 = q, system status
Figure BDA0003187648080000134
x 3 = ν order output signal y of single-joint mechanical arm control system = q, then the nonlinear model of the single-joint mechanical arm system can be expressed as follows
Figure BDA0003187648080000141
Wherein f is 1 (x)＝0，g 1 (x 1 )＝1，f 2 (x)＝-10sin(x 1 )-x 2 ，g 2 (x 2 )＝1，f 3 (x)＝-0.2x 2 -x 3 ，g 3 (x 3 )＝1；f 1 (x)，f 2 (x)，f 3 (x)，g 1 (x 1 )，g 2 (x 2 ) And g 3 (x 3 ) Are all thatIn the definition domain
Figure BDA0003187648080000142
A non-linear function with fully smooth inner surface and satisfying g 1 (x 1 )≠0，g 2 (x 2 ) Not equal to 0 and g 3 (x 3 )≠0；
Step 1.3, considering that some state variables in the system may be unobservable, a fuzzy state observer needs to be designed to estimate these unobservable states. Thus, the above system can be expressed as the following state space equation:
Figure BDA0003187648080000143
in the formula
Figure BDA0003187648080000144
K＝(k 1 ,k 2 ,k 3 ) T ,B i ＝(0,1,0) T ,B＝(0,0,1) T C = (1,0,0). A is a strict Hurwitz matrix, and by choosing the appropriate K, there is a positive definite matrix Q = Q T ＞0，P＝P T > 0 satisfies A T P+PA＝-Q
Step 1.4, setting the corresponding state observer as follows:
Figure BDA0003187648080000145
in the formula
Figure BDA0003187648080000146
Each represents x = (x) 1 ,x 2 ,x 3 ) T ，f i (x) An estimate of (d).
Based on the fuzzy logic rules, we can get:
Figure BDA0003187648080000151
Figure BDA0003187648080000152
in the formula of i Which represents the minimum approximation error, is,
Figure BDA0003187648080000153
represents the optimal weight vector, if any
Figure BDA0003187648080000154
Satisfy the requirement of
Figure BDA0003187648080000155
The observation error can be expressed as
Figure BDA0003187648080000156
Wherein δ = (δ) 1 ,δ 2 ,δ 3 ) T ，
Figure BDA0003187648080000157
Step 1.5, constructing a corresponding Lyapunov function as:
Figure BDA0003187648080000158
derivation of this can yield:
Figure BDA0003187648080000159
in view of the young's inequality and the fuzzy basis function
Figure BDA00031876480800001510
The following can be obtained:
Figure BDA00031876480800001511
wherein
Figure BDA00031876480800001512
Bringing the inequality into
Figure BDA00031876480800001513
The following can be obtained:
Figure BDA00031876480800001514
in the formula
Figure BDA00031876480800001515
Step 2, according to the state space model of the single-joint mechanical arm system established in the step 1, introducing a barrier function to solve the problem of multi-target constraint, constructing a first Lyapunov function, and setting a corresponding virtual control law and a corresponding parameter self-adaptive law, wherein the method specifically comprises the following steps:
step 2.1, the barrier function is designed as follows:
Figure BDA0003187648080000161
wherein the content of the first and second substances,
Figure BDA0003187648080000162
m i (i = 1.... N) represents a weighting coefficient; the appropriate weighting coefficients are chosen to ensure that the overall objective function is constrained within a specified range. Since I is x 1 In the open set Ω, the initial value I (0) is in the domain. If it is used
Figure BDA0003187648080000163
Or
Figure BDA0003187648080000164
ξ → ∞. Briefly, theI also follows the constraint as long as ξ is guaranteed to be bounded.
Thus, the constraint problem of satisfying the objective function can translate into guaranteeing the boundedness of ξ.
Derivation of I yields:
Figure BDA0003187648080000165
wherein
Figure BDA0003187648080000166
Subsequently, the air conditioner is operated to,
Figure BDA0003187648080000167
can be rewritten as:
Figure BDA0003187648080000168
Figure BDA0003187648080000169
wherein
Figure BDA00031876480800001610
And
Figure BDA00031876480800001611
at the same time, χ can be deduced 1,0 Not equal to 0. If x 1,1 Not equal to 0, then x 1,1 ＝χ 1,0 sign(χ 1,0 )。
As long as I = x 1 Then the multi-objective constraint problem is transformed into an output constraint, which is the content of the constraint studied in engineering.
Step 2.2, the following coordinate transformation is defined:
z 1 (t)＝ξ-y d ,
Figure BDA0003187648080000171
Figure BDA0003187648080000172
Figure BDA0003187648080000173
Figure BDA0003187648080000174
where xi is the barrier function, z i As systematic state error, y d As a reference signal, the reference signal is,
Figure BDA0003187648080000175
to compensate for error signals, η i Is an error compensation signal.
Step 2.3, the first order command filter is required to be introduced in the application
Figure BDA0003187648080000176
To overcome the defect of the existing framework based on the self-adaptive backstepping method that the virtual control signal alpha is subjected to i To reduce the corresponding computational burden. However, the filtering error caused by the first order command filter is mostly ignored in the existing results
Figure BDA0003187648080000177
When we introduce the following error compensation mechanism to solve the filtering error
Figure BDA0003187648080000178
In which
Figure BDA0003187648080000179
Is a first order filter output signal, alpha i Input signal, beta, representing a first order filter i > 0 is a time constant.
Figure BDA00031876480800001710
Wherein eta i (0)＝0,χ j,1 ＝1(j＝2,...,n)，k i1 ＞0,k i2 > 0 is a design parameter.
Figure BDA00031876480800001711
Error compensation signal incorporating the above
Figure BDA00031876480800001712
The following can be obtained:
Figure BDA00031876480800001713
constructing the Lyapunov function
Figure BDA0003187648080000181
Wherein
Figure BDA0003187648080000182
In order to estimate the error for the parameter,
Figure BDA0003187648080000183
at the same time to V 1 Derivation can be obtained:
the Lyapunov function is selected according to the Lyapunov function selected by the similar reference:
Figure BDA0003187648080000184
using fuzzy basis functions
Figure BDA0003187648080000185
And can be obtained by processing the Young inequality:
Figure BDA00031876480800001813
Figure BDA0003187648080000186
where τ > 0, the above formula can be substituted:
Figure BDA0003187648080000187
law of virtual control
Figure BDA0003187648080000188
And law of parameter adaptation
Figure BDA0003187648080000189
And
Figure BDA00031876480800001810
wherein k is 11 ,k 12 ,τ,σ 1 ,c 1 ,r 1 ,
Figure BDA00031876480800001811
Are all normal numbers;
the virtual control law and the parameter adaptive law are brought into being available:
Figure BDA00031876480800001812
wherein
Figure BDA0003187648080000191
Step 3, constructing a second Lyapunov function according to the state space model of the single-joint mechanical arm system established in the step 1, and setting a corresponding virtual control law and a corresponding parameter self-adaptive law, wherein the method specifically comprises the following steps:
combining the state space model of the single-joint mechanical arm system in the step 2 with coordinate transformation:
Figure BDA0003187648080000192
wherein
Figure BDA0003187648080000193
And introducing an error compensation signal to solve the influence of filtering errors:
Figure BDA0003187648080000194
constructing a second Lyapunov function for ensuring the stability of the single-joint mechanical arm system:
Figure BDA0003187648080000195
the derivation of which is:
Figure BDA0003187648080000196
using fuzzy basis functions as in step 2
Figure BDA0003187648080000197
And young inequality treatment can obtain:
Figure BDA0003187648080000198
Figure BDA0003187648080000199
the corresponding formula would be substituted:
Figure BDA0003187648080000201
law of virtual control
Figure BDA0003187648080000202
And law of parameter adaptation
Figure BDA0003187648080000203
And
Figure BDA0003187648080000204
wherein k is 21 ，k 22 ，τ，σ 2 ，c 2 ，r 2 ，
Figure BDA0003187648080000205
Are all normal numbers;
the virtual control law and the parameter adaptive law are brought into being available:
Figure BDA0003187648080000206
wherein
Figure BDA0003187648080000207
Step 4, constructing a third Lyapunov function according to the state space model of the single-joint mechanical arm system established in the step 1, and setting a corresponding virtual control law and a corresponding parameter self-adaptive law, wherein the method specifically comprises the following steps:
the state space model and the coordinate transformation of the single-joint mechanical arm system combined with the steps can obtain:
Figure BDA0003187648080000208
and introducing an error compensation signal to solve the influence of filtering errors:
Figure BDA0003187648080000209
constructing a third Lyapunov function for ensuring the stability of the single-joint mechanical arm system:
Figure BDA00031876480800002010
the derivation of which is:
Figure BDA0003187648080000211
using fuzzy basis functions as above
Figure BDA0003187648080000212
And the young inequality can be given as:
Figure BDA0003187648080000213
before setting the actual event trigger controller u, the following virtual control law α is set in the present application 4 And law of parameter adaptation
Figure BDA0003187648080000214
Figure BDA0003187648080000215
Figure BDA0003187648080000216
Step 5, on the basis of the steps, introducing an event triggering strategy to reduce the communication burden, so that the single-joint mechanical arm system meets the actual fixed time stability condition, namely completing the track tracking control of the single-joint mechanical arm system, and specifically comprising the following steps of:
by introducing an event trigger control strategy based on a relative threshold value, the corresponding communication burden and the waste of communication resources are reduced.
The relative threshold based event-triggered control strategy is described in detail below:
Figure BDA0003187648080000217
Figure BDA0003187648080000218
t k+1 ＝inf{t∈R||P(t)|≥τ|u(t)|+μ 2 }
defining an event trigger error P (t) = v (t) -u (t), 0 < τ < 1, ρ, μ 1 ,μ 2 Are all normal numbers and satisfy
Figure BDA0003187648080000219
t k ,k∈z + Representing the input update time. It is noted that at time t e t k ,t k+1 ) U can be regarded as v (t) k )，
Each time t k+1 ＝inf{t∈R||P(t)|≥τ|u(t)|+μ 2 When triggered, the time instant will be marked as t k+1 Actual control input u (t) k+1 ) Will be applied to the system. Therefore, we can find the parameter l satisfying the following equation 1 (t),l 2 (t)：
v(t)＝(1+l 1 (t)τ)u+l 2 (t)μ 2
Wherein | l 1 (t)|≤1，|l 2 (t) | ≦ 1, thus giving the controller:
Figure BDA0003187648080000228
at interval time t k ,t k+1 In the method, based on the event trigger control strategy, | v (t) -u (t) | < tau | u (t) | + mu can be obtained 2 The controller u is set as
Figure BDA0003187648080000221
Wherein
Figure BDA0003187648080000222
The following can be obtained:
Figure BDA0003187648080000223
since 0 < 1+l 1 (t) τ < 1+ τ and
Figure BDA0003187648080000224
can obtain the product
Figure BDA0003187648080000225
Bringing into availability:
Figure BDA0003187648080000226
based on the theory 1:
Figure BDA0003187648080000227
the following can be obtained:
Figure BDA0003187648080000231
wherein M is 3 ＝M 2 +0.557ρ；
Definition of
Figure BDA0003187648080000232
Based on the theory 2:
Figure BDA0003187648080000233
Figure BDA0003187648080000234
Figure BDA0003187648080000235
based on the theory 3: h n ∈R,i＝1,...,n,κ∈[0,1]
(|H 1 |+…+|H n |) κ ≤|H 1 | κ +…+|H n | κ
Figure BDA0003187648080000236
In view of
Figure BDA0003187648080000237
Figure BDA0003187648080000238
Figure BDA0003187648080000239
Bringing the above two inequalities into
Figure BDA00031876480800002310
The following can be obtained:
Figure BDA0003187648080000241
wherein
Figure BDA0003187648080000242
Definition of
Figure BDA0003187648080000243
Figure BDA0003187648080000244
And (4) introduction: x is a radical of a fluorine atom 1 ，y 2 Represents an arbitrary variable, k 1 ，k 2 And B represents an arbitrary constant, and B represents,
Figure BDA0003187648080000245
Figure BDA0003187648080000246
Figure BDA0003187648080000247
Figure BDA0003187648080000248
here τ 1 ＝0.11；
Bringing the above into
Figure BDA0003187648080000251
Can obtain
Figure BDA0003187648080000252
Wherein
Figure BDA0003187648080000253
Based on
Figure BDA0003187648080000254
And destructively processing by the following young inequality:
Figure BDA0003187648080000255
Figure BDA0003187648080000256
lyapunov differential function
Figure BDA0003187648080000257
Can be expressed as:
Figure BDA0003187648080000258
in the formula
Figure BDA0003187648080000259
Figure BDA00031876480800002510
Definition of
Figure BDA0003187648080000261
According to
Figure BDA0003187648080000262
And applying lemmas 2 and 3 as above, the above formula can be converted to:
Figure BDA0003187648080000263
at this time, it is assumed that there is an unknown constant
Figure BDA0003187648080000264
Satisfy the requirement of
Figure BDA0003187648080000265
The following two cases were analyzed:
case 1: if it is not
Figure BDA0003187648080000266
Figure BDA0003187648080000267
Figure BDA0003187648080000268
Thus can obtain
Figure BDA0003187648080000269
Case 2: if it is not
Figure BDA00031876480800002610
Figure BDA00031876480800002611
Figure BDA00031876480800002612
Is provided with
Figure BDA00031876480800002613
Summarizing the above two cases can be found:
Figure BDA0003187648080000271
wherein
Figure BDA0003187648080000272
Figure BDA0003187648080000273
According to the theory 5: if V (x) is a positive definite function and has the form
Figure BDA0003187648080000274
In the formula 1 ,φ 2 Each of α, β, γ represents a normal number, and satisfies α γ ∈ (0,1), β γ ∈ (1, ∞), ρ > 0.
It can be demonstrated that the origin of the system has reached actual fixed time stability (the advantage of actual fixed time stability as chosen herein has the advantage that the convergence time can be predicted normally regardless of the initial conditions, as opposed to asymptotic stability or finite time stability).
Referring to the existing literature, the parameters were selected as follows
Figure BDA0003187648080000275
β =2 and γ =1 are more convenient for practical design.
Therefore, the single-joint mechanical arm system can meet the actual fixed time stability condition.
The design goal of the present application is to design the controller u such that the output signal y can be constrained to a limited range (k) c1 ,k c2 ) Internal simultaneous tracking reference signal y d And ensures the tracking error z 1 The convergence to the small neighborhood range of zero in a fixed time interval effectively reduces the calculated amount and accelerates the convergence speed; the tracking tracks of the output signal, the observation signal and the reference signal of the single-joint mechanical arm are shown in fig. 3. A schematic of the tracking error for a single joint robotic arm is shown in fig. 4.
The track tracking control research is carried out by taking a typical repeated motion nonlinear system such as a single-joint mechanical arm as an object, and compared with a finite time algorithm, the track tracking control research has higher convergence rate; compared with the general self-adaptive backstepping control, the method can reduce the communication burden and the calculated amount, so that the method has higher engineering practical value for the research of the single-joint mechanical arm.
The previous description of the disclosed embodiments is provided to enable any person skilled in the art to make or use the present invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the generic principles defined herein may be applied to other embodiments without departing from the spirit or scope of the invention. Thus, the present invention is not intended to be limited to the embodiments shown herein but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims (1)

1. The track tracking control method of the single-joint mechanical arm system under multi-target constraint is characterized by comprising the following steps of:
step 1, establishing a state space model of a single-joint mechanical arm according to a mathematical model of a single-joint mechanical arm system, constructing a corresponding state observer to estimate an unmeasured state, and finally performing luggage Jaconov stability analysis by referring to an observation error system;
the step 1 specifically comprises the following steps:
step 1.1, firstly, according to a system structure diagram of the single-joint mechanical arm, establishing a nonlinear mathematical model of the single-joint mechanical arm as follows:
Figure FDA0003818195350000011
Figure FDA0003818195350000012
wherein
Figure FDA0003818195350000013
q represents the acceleration, velocity and position of the stick, respectively, v represents the torque induced by the power subsystem, u represents the control input, D =1.5kg m 2 Representing mechanical inertia, B =1Nm s/rad representing the viscous friction coefficient at the joint, H =1 Ω representing the armature resistance, M = H representing the armature inductance, L =0.2Nm/a representing the back emf coefficient;
step 1.2, define the system state variable x 1 = q, system status
Figure FDA0003818195350000014
x 3 And = ν, and let the output signal y of the single-joint mechanical arm control system = q, the nonlinear model of the single-joint mechanical arm system can be represented as follows:
Figure FDA0003818195350000015
wherein f is 1 (x)＝0，g 1 (x 1 )＝1，f 2 (x)＝-10sin(x 1 )-x 2 ，g 2 (x 2 )＝1，f 3 (x)＝-0.2x 2 -x 3 ，g 3 (x 3 )＝1；
f 1 (x)，f 2 (x)，f 3 (x)，g 1 (x 1 )，g 2 (x 2 ) And g 3 (x 3 ) Are all in the domain of definition
Figure FDA0003818195350000016
A non-linear function with fully smooth inner surface and satisfying g 1 (x 1 )≠0，g 2 (x 2 ) Not equal to 0 and g 3 (x 3 )≠0；
Step 1.3, representing the nonlinear model of the single-joint mechanical arm system as a state space model as follows:
Figure FDA0003818195350000021
in the formula (I), the compound is shown in the specification,
Figure FDA0003818195350000022
K＝(k 1 ,k 2 ,k 3 ) T ,B i ＝(0,1,0) T ,B＝(0,0,1) T c = (1,0,0); a is a strict Hurwitz matrix, and by choosing the appropriate K, there is a positive definite matrix Q = Q T ＞0，P＝P T Is greater than 0 and satisfies A T P+PA＝-Q；
Step 1.4, setting the corresponding state observer as follows:
Figure FDA0003818195350000023
in the formula
Figure FDA0003818195350000024
Figure FDA0003818195350000025
Each represents x = (x) 1 ,x 2 ,x 3 ) T ，f i (x) An estimated value of (d);
based on fuzzy logic rules, we can get:
Figure FDA0003818195350000026
Figure FDA0003818195350000027
in the formula of i Which represents the minimum approximation error, is,
Figure FDA0003818195350000028
represents the optimal weight vector, if any
Figure FDA0003818195350000029
Satisfy the requirements of
Figure FDA00038181953500000210
The observation error can be expressed as
Figure FDA00038181953500000211
Wherein δ = (δ) 1 ,δ 2 ,δ 3 ) T ，
Figure FDA00038181953500000212
Step 1.5, constructing a corresponding Lyapunov function as:
Figure FDA00038181953500000213
derivation of this can yield:
Figure FDA0003818195350000031
in view of the Young's inequality and fuzzy basis functions
Figure FDA0003818195350000032
The following can be obtained:
Figure FDA0003818195350000033
wherein
Figure FDA0003818195350000034
Bringing the inequality of the above into
Figure FDA0003818195350000035
The following can be obtained:
Figure FDA0003818195350000036
in the formula, λ 0 ＝λ min (Q)-1,
Figure FDA0003818195350000037
Step 2, according to the state space model of the single-joint mechanical arm system established in the step 1, introducing a barrier function to solve the problem of multi-target constraint, constructing a first Lyapunov function, and setting a corresponding virtual control law and a parameter self-adaptation law;
the step 2 specifically comprises the following steps:
step 2.1, the barrier function is designed as follows:
Figure FDA0003818195350000038
wherein the content of the first and second substances,
Figure FDA0003818195350000039
m i i = 1.. And n denotes a weighting coefficient;
step 2.2, the following coordinate transformation is defined:
z 1 (t)＝ξ-y d ,
Figure FDA0003818195350000041
Figure FDA0003818195350000042
Figure FDA0003818195350000043
Figure FDA0003818195350000044
where xi is the barrier function, z i As systematic state error, y d As a reference signal, to be used as a reference signal,
Figure FDA0003818195350000045
to compensate for error signals, eta i An error compensation signal;
step 2.3, the following error compensation mechanism is introduced to solve the filteringWave error
Figure FDA0003818195350000046
The influence of (a):
Figure FDA0003818195350000047
wherein
Figure FDA0003818195350000048
Is the output signal of a first-order filter, alpha i Input signal, beta, representing a first order filter i > 0 is a time constant; eta i (0)＝0,χ j,1 ＝1(j＝2,...,n)，k i1 ＞0,k i2 > 0 is a design parameter;
Figure FDA0003818195350000049
error compensation signal incorporating the above
Figure FDA00038181953500000410
The following can be obtained:
Figure FDA00038181953500000411
constructing the Lyapunov function
Figure FDA00038181953500000412
Wherein
Figure FDA00038181953500000413
In order to estimate the error for the parameter,
Figure FDA00038181953500000414
at the same time to V 1 The derivation can be:
Figure FDA0003818195350000051
using fuzzy basis functions
Figure FDA0003818195350000052
And can be obtained by processing the Young inequality:
Figure FDA0003818195350000053
Figure FDA0003818195350000054
where τ > 0, the above formula can be substituted:
Figure FDA0003818195350000055
law of virtual control
Figure FDA0003818195350000056
And law of parameter adaptation
Figure FDA0003818195350000057
And
Figure FDA0003818195350000058
wherein k is 11 ,k 12 ,τ,σ 1 ,c 1 ,r 1 ,
Figure FDA0003818195350000059
Are all normal numbers;
the virtual control law and the parameter adaptive law are brought into being available:
Figure FDA00038181953500000510
wherein
Figure FDA00038181953500000511
Step 3, constructing a second Lyapunov function according to the state space model of the single-joint mechanical arm system established in the step 1, and setting a corresponding virtual control law and a corresponding parameter self-adaptive law;
the step 3 specifically comprises the following steps:
combining the state space model of the single-joint mechanical arm system in the step 2 and the coordinate transformation, the following can be obtained:
Figure FDA0003818195350000061
wherein
Figure FDA0003818195350000062
And introducing an error compensation signal to solve the influence of filtering errors:
Figure FDA0003818195350000063
constructing a second Lyapunov function for ensuring the stability of the single-joint mechanical arm system:
Figure FDA0003818195350000064
the derivation of which is:
Figure FDA0003818195350000065
using fuzzy basis functions
Figure FDA0003818195350000066
And young inequality treatment can obtain:
Figure FDA0003818195350000067
Figure FDA0003818195350000068
replacing the corresponding formula can be:
Figure FDA0003818195350000069
law of virtual control
Figure FDA0003818195350000071
And law of parameter adaptation
Figure FDA0003818195350000072
And
Figure FDA0003818195350000073
wherein k is 21 ，k 22 ，τ，σ 2 ，c 2 ，r 2 ，
Figure FDA0003818195350000074
Are all normal numbers;
the virtual control law and the parameter adaptive law are brought into being available:
Figure FDA0003818195350000075
wherein
Figure FDA0003818195350000076
Step 4, constructing a third Lyapunov function according to the state space model of the single-joint mechanical arm system established in the step 1, and setting a corresponding virtual control law and a corresponding parameter self-adaptive law;
the step 4 specifically comprises the following steps:
the state space model and the coordinate transformation of the single-joint mechanical arm system combined with the steps can obtain:
Figure FDA0003818195350000077
and introducing an error compensation signal to solve the influence of filtering errors:
Figure FDA0003818195350000078
constructing a third Lyapunov function for ensuring stability of the single-joint mechanical arm system
Figure FDA0003818195350000079
The derivation of which is:
Figure FDA0003818195350000081
using fuzzy basis functions as above
Figure FDA0003818195350000082
And the young inequality can be given as:
Figure FDA0003818195350000083
before setting the actual event-triggered controller u, the virtual control law α is set as follows 4 And law of parameter adaptation
Figure FDA0003818195350000084
Figure FDA0003818195350000085
Figure FDA0003818195350000086
Step 5, introducing an event trigger strategy to reduce the communication burden on the basis of the steps, so that the single-joint mechanical arm system meets the actual fixed time stability condition, and the track tracking control of the single-joint mechanical arm system is completed;
the step 5 specifically comprises the following steps:
defining the event trigger error as P (t) = v (t) -u (t)
Wherein, the first and the second end of the pipe are connected with each other,
Figure FDA0003818195350000087
ρ,μ 1 ,μ 2 are all normal numbers and satisfy
Figure FDA0003818195350000088
t k ,k∈z + Representing an input update time;
at interval time t k ,t k+1 ) In the method, | v (t) -u (t) | < τ | u (t) | + μ is obtained based on the event-triggered control strategy 2 The controller u is set as
Figure FDA0003818195350000089
Wherein
Figure FDA00038181953500000810
The following can be obtained:
Figure FDA0003818195350000091
since 0 < 1+l 1 (t) τ < 1+ τ and
Figure FDA0003818195350000092
can obtain the product
Figure FDA0003818195350000093
Bringing into availability:
Figure FDA0003818195350000094
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