CN114851198A - Consistent tracking fixed time stability control method for multi-single-link mechanical arm - Google Patents

Consistent tracking fixed time stability control method for multi-single-link mechanical arm Download PDF

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CN114851198A
CN114851198A CN202210536923.3A CN202210536923A CN114851198A CN 114851198 A CN114851198 A CN 114851198A CN 202210536923 A CN202210536923 A CN 202210536923A CN 114851198 A CN114851198 A CN 114851198A
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mechanical arm
error
follows
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CN114851198B (en
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王建晖
王晨
杜泳萍
吴宇深
李咏华
张春良
朱厚耀
刘嘉睿
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Guangzhou University
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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
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    • B25J9/1602—Programme controls characterised by the control system, structure, architecture
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    • Y02P90/00—Enabling technologies with a potential contribution to greenhouse gas [GHG] emissions mitigation
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Abstract

The invention discloses a consistent tracking fixed time stability control method of a multi-single-link mechanical arm, which comprises the following steps: modeling the single-link mechanical arm to obtain a state equation; defining the consistency tracking error of the ith single-link mechanical arm and designing a first virtual control law alpha i,1 (ii) a Designing a relative threshold event trigger mechanism; design a second virtual law α i,2 And law of adaptation
Figure DDA0003648687520000011
And
Figure DDA0003648687520000012
estimating unknown system parameters on line; and carrying out simulation experiments on the algorithm based on a Matlab experiment platform. According to the invention, through a relative threshold event trigger mechanism in the controller, on-line dynamic compensation of an input dead zone can be realized on the premise of ensuring the control precision, and meanwhile, the updating frequency of input signals of each mechanical arm is reduced; multiple single connectionThe rod mechanical arm system can realize the convergence of synchronous errors in fixed time under different system initial states, the convergence time is irrelevant to the system initial state, and higher convergence speed and better tracking accuracy are obtained by selecting proper parameters.

Description

Consistent tracking fixed time stability control method for multi-single-link mechanical arm
Technical Field
The invention relates to the technical field of intelligent control of mechanical arms, in particular to a consistent tracking fixed time stability control method for a multi-single-connecting-rod type mechanical arm.
Background
With the continuous expansion of the application range of the mechanical arm, the mutually independent mechanical arms show more and more limitations, and the multiple mechanical arms can be operated flexibly in a cooperative mode, have strong robustness and can complete complex tasks. The mechanical arms are connected through a topological network, and communication among the mechanical arms is achieved. High-frequency communication is needed among subsystems to ensure the stability of the whole system. However, the communication resources of the system are limited, and the conventional periodic sampling control method may have the problem of network congestion. In addition, there is often an input dead zone in the robotic arm system that affects the stability of the system. Therefore, it is of great importance to study the event-triggered consistency tracking problem for a set of single-link robotic arms with input dead zone constraints.
The convergence speed is an important index of system performance, the existing finite time control method can realize the finite time stabilization of the system, but the convergence time of the system is related to the initial state of the system. In practical applications, the system state is often not measurable. Therefore, this method cannot calculate the settling time in advance. The invention designs the controller based on the fixed time stabilization theory, can realize the rapid finite time stabilization of the system, and the convergence time is irrelevant to the initial state of the system.
Most of the existing technical schemes do not consider that the communication resources of the system are limited, and a large amount of communication resources are occupied to maintain the stability of the system and realize the compensation of the input dead zone. The invention designs an event trigger mechanism, which can reduce the update frequency of control signals, realize the self-adaptive compensation of input dead zones and relieve the communication pressure of a system to a certain extent.
Disclosure of Invention
The invention aims to provide a consistent tracking fixed time stability control method for a plurality of single-link mechanical arms, which solves the problems in the background technology by designing a new relative threshold event trigger mechanism.
In order to achieve the purpose, the invention provides the following technical scheme:
a consistent tracking fixed time stability control method for a multi-single-link mechanical arm comprises the following steps:
s01, modeling the single-link mechanical arm to obtain a state equation;
s02, defining the consistency tracking error of the ith single-link mechanical arm and designing a first virtual control law alpha i,1 ；
S03, designing a relative threshold event trigger mechanism;
s04, designing a second virtual law alpha i,2 And law of adaptation
Figure BDA0003648687500000021
And
Figure BDA0003648687500000022
estimating unknown system parameters on line;
and S05, carrying out simulation experiments on the algorithm based on the Matlab experiment platform.
Preferably, the step S01 is specifically as follows:
without loss of generality, the mathematical model of the ith single link arm in the follower is set up as follows:
Figure BDA0003648687500000023
wherein,t represents time, q i (t) is the angle of the joint,
Figure BDA0003648687500000024
is the angular velocity of the joint or joints,
Figure BDA0003648687500000025
is angular acceleration of the joint, J i Is moment of inertia, B i Is the coefficient of friction damping, m i Is the connecting rod mass, g is the gravitational acceleration,/ i Is the length of the connecting rod, u i (t) is an input torque.
Due to the structural characteristics of the mechanical structure, the mechanical arm system often has an input dead zone, which has a great influence on the system performance. The mathematical model for setting the input dead zone is as follows:
D i (u i )＝h i u i +g i (2)
wherein ,ui Represents a control input;
Figure BDA0003648687500000026
h i,1 、h i,2 、g i,1 and gi,2 Are all constant, and h i,1 ＞0、h i,2 ＞0、g i,1＜0 and gi,2 ＞0。
Coordinate transformation of (1) it can result in:
Figure BDA0003648687500000027
wherein ,xi,1 ＝q i (t)，
Figure BDA0003648687500000028
Each represents the system state of the i (i ═ 1, 2.., N) th follower;
Figure BDA0003648687500000029
denotes x i,1 A derivative of (a);
Figure BDA00036486875000000210
denotes x i.2 A derivative of (a); y is i A system output representing the i (i ═ 1, 2.., N) th follower;
Figure BDA0003648687500000031
indicating a system uncertainty portion.
In order to facilitate the description of the communication relationship between the mechanical arms in the topological graph, relevant knowledge of algebraic graph theory needs to be introduced. Drawing (A)
Figure BDA0003648687500000032
Directed communication topology diagram representing multi-arm system, wherein each node in the diagram corresponds to one arm, and Ω ═ Ω { Ω [ Ω ] 1 ,Ω 2 ,...,Ω N Denotes the set of N nodes, the set of edges between nodes is
Figure BDA0003648687500000033
The edge from node i to node j is defined as an ordered pair
Figure BDA0003648687500000034
Indicating that the mechanical arm i can receive the information of the mechanical arm j, and calling the node i to be adjacent to the node j to define
Figure BDA0003648687500000035
Is the set of adjacent edges of agent i. A ═ a i,j ]∈R N×N Represents a adjacency matrix if
Figure BDA0003648687500000036
Then a i,j Is greater than 0; otherwise a i,j 0. The degree of entry of the node i is
Figure BDA0003648687500000037
Definition of
Figure BDA0003648687500000038
Figure BDA0003648687500000039
Is an in-degree diagonal matrixThen, then
Figure BDA00036486875000000310
Is a laplacian matrix.
To deal with unknown non-linear functions in the system, radial basis function neural networks are employed. In tight set omega epsilon R n Any continuous function defined above
Figure BDA00036486875000000311
Can be approximated by a neural network and can be expressed as:
Figure BDA00036486875000000312
wherein ,
Figure BDA00036486875000000313
is an ideal unknown weight vector, and
Figure BDA00036486875000000314
q is a positive integer;
Figure BDA00036486875000000315
to represent
Figure BDA00036486875000000316
Transposing; Φ (X) is a vector of basis functions, and Φ (X) ═ Φ 1 (X),Φ 2 (X),...,Φ q (X)] T (ii) a q is the node number of the neural network, and q is more than 1; x represents an input vector, and X ═ X 1 ,x 2 ,...,x q ] T ∈R q (ii) a Mu (X) is an approximation error and satisfies
Figure BDA00036486875000000317
Figure BDA00036486875000000318
and Φi The definition of (X) is as follows:
Figure BDA00036486875000000319
Figure BDA00036486875000000320
wherein ,Wi ＝[W 1 ,W 2 ,...,W q ] T ∈R q Representing a weight vector;
Figure BDA00036486875000000321
and
Figure BDA00036486875000000322
are respectively a Gaussian function phi i The center and width of (X).
Preferably, the step S02 is as follows:
first, the consistency tracking error of the ith robot arm is defined by the knowledge of graph theory:
Figure BDA0003648687500000041
wherein ,zi,1 For synchronization error, z i,2 Is a virtual control error, α i,1 Is the virtual control law, y d A signal is output for the leader. b i (i ═ 1, 2.., N) denotes the information transmission coefficient from the leader to the i-th follower. b i For positive numbers, if there is a transmission of information between the leader and the ith follower, b i Is greater than 0. Otherwise b i 0. As shown in the formula (7), the synchronization error z i,1 Weighted parameter a ij and bi The influence of (c). Therefore, a directed graph
Figure BDA00036486875000000417
Structural influence of (a) z i,1 . Further according to d i and bi Definition of (d) i +b i Is strictly positive.
Preferably, the step S02 specifically includes:
s021, processing unknown combined parts by adopting radial basis function neural network
Figure BDA0003648687500000042
Introducing an unknown positive parameter
Figure BDA0003648687500000043
Wherein | · | | represents a two-norm;
Figure BDA0003648687500000044
denotes y d The first derivative of (a); x i,1 Represents an input vector, an
Figure BDA0003648687500000045
x j,1 and xj,2 Represents the system state of the jth (j ═ 1, 2.., N) follower; and the parameter theta i,1 Can pass through
Figure BDA0003648687500000046
Estimate, i.e.
Figure BDA0003648687500000047
Is a parameter theta i,1 Then the final estimation error can be defined as
Figure BDA0003648687500000048
Thus, approximation processing
Figure BDA0003648687500000049
The expression of (a) is as follows:
Figure BDA00036486875000000410
wherein ,
Figure BDA00036486875000000411
is an ideal unknown weight vector, and
Figure BDA00036486875000000412
Figure BDA00036486875000000413
to represent
Figure BDA00036486875000000414
Transposing; phi (X) is a vector of basis functions, and phi i,1 (X i,1 )＝[Φ 1,1 (X i,1 ),Φ 2,1 (X i,1 ),...,Φ q,1 (X i,1 )] T (ii) a q is the node number of the neural network, and q is more than 1; mu.s i,1 (X i,1 ) Represents an approximation error, satisfies
Figure BDA00036486875000000415
And is
Figure BDA00036486875000000416
S022, first virtual control law α i,1 The design is as follows:
Figure BDA0003648687500000051
Figure BDA0003648687500000052
wherein ,ci,1 、r i,1 、a i,1 and εi,1 Are all constant, and c i,1 ＞0，r i,1 ＞0，a i,1 ＞0，ε i,1 ＞0。
Preferably, the step S03 is specifically as follows:
in order to save communication resources, a relative threshold event triggering mechanism is designed. Control signal omega i (t) is defined as follows:
Figure BDA0003648687500000053
only when pre-designed triggeringCondition | e i (t)|≥ρ i |u i (t)|+ο i When true, controls the input signal u i (t) is updated, the expression is as follows:
Figure BDA0003648687500000054
wherein δi 、ο i 、ρ i 、
Figure BDA0003648687500000055
Are all positive design parameters and satisfy delta i ＞0，0＜ρ i ＜1，ο＞0，
Figure BDA0003648687500000056
k is an integer. inf {. } represents an infimum bound; t is t i,k The kth trigger time of the ith agent; t is t i,k+1 The (k + 1) th trigger time of the ith agent; e.g. of the type i (t) denotes measurement error, and e i (t)＝ω i (t)-u i (t)。
Preferably, because the system model has an uncertain part, the uncertain part of the second-order nonlinear system model is approximated through a neural network; designing a virtual control law according to the virtual control error, and determining adaptive parameters, wherein the step S04 specifically includes:
determining uncertain combination parts in system model by radial basis function neural network
Figure BDA0003648687500000057
Figure BDA0003648687500000058
Denotes y d The second derivative of (a); x i,2 Represents an input vector, an
Figure BDA0003648687500000059
Introducing an unknown positive parameter
Figure BDA00036486875000000510
Wherein | · | | represents a two-norm; and the parameter theta i,2 Can pass through
Figure BDA0003648687500000061
Estimate, i.e.
Figure BDA0003648687500000062
Is a parameter theta i,2 Then the final estimation error can be defined as
Figure BDA0003648687500000063
Thus, approximation processing
Figure BDA0003648687500000064
The expression of (a) is as follows:
Figure BDA0003648687500000065
wherein ,
Figure BDA0003648687500000066
is an ideal unknown weight vector, and
Figure BDA0003648687500000067
Figure BDA0003648687500000068
to represent
Figure BDA0003648687500000069
Transposing;
Figure BDA00036486875000000610
is a vector of basis functions, and phi i,2 (X i,2 )＝[Φ 1,2 (X i,2 ),Φ 2,2 (X i,2 ),...,Φ q,2 (X i,2 )] T (ii) a q is the node number of the neural network, and q is more than 1; mu.s i,2 (X i,2 ) Represents an approximation error, satisfies
Figure BDA00036486875000000611
And is
Figure BDA00036486875000000612
S042 according to the second error variable z i,2 In the process, the unknown uncertain part is obtained by solving the radial basis function neural network, and a second virtual control law alpha is designed by utilizing a backstepping design method and the Lyapunov function i,2 While generating adaptive parameters
Figure BDA00036486875000000613
And
Figure BDA00036486875000000614
in which a virtual control law alpha is established i,2 、
Figure BDA00036486875000000615
Law of control
Figure BDA00036486875000000616
And adaptive parameters
Figure BDA00036486875000000617
Figure BDA00036486875000000618
Satisfies the following calculation formula:
Figure BDA00036486875000000619
Figure BDA00036486875000000620
Figure BDA00036486875000000621
Figure BDA00036486875000000622
Figure BDA00036486875000000623
wherein
Figure BDA00036486875000000624
Represents phi i Transpose of (J) i 、ε i,2 、c i,2 、r i,2 、a i,2 、ξ i 、σ i 、l i,2 、η i And ζ i Are all positive design parameters. H i ＝[1/h i ,-g i /h i ] T ，H i T Represents H i Transpose of (Q) i ＝[α i,2 ,1] T ，α i,2 Representing a second virtual control law; since in practical cases, H i Are difficult to obtain and design
Figure BDA00036486875000000625
To estimate H i Defining an estimation error
Figure BDA00036486875000000626
And is
Figure BDA0003648687500000071
wherein
Figure BDA0003648687500000072
And
Figure BDA0003648687500000073
are respectively estimated as
Figure BDA0003648687500000074
And
Figure BDA0003648687500000075
Figure BDA0003648687500000076
multiplication by multiplication
Figure BDA0003648687500000077
Compared with the prior art, the invention has the beneficial effects that:
1. according to the consistent tracking fixed time stability control method for the multi-single-link mechanical arm, in order to reduce communication burden among the mechanical arms, a new relative threshold event trigger mechanism is designed in a controller, on-line dynamic compensation of an input dead zone can be achieved on the premise that control precision is guaranteed, and meanwhile the updating frequency of input signals of all the mechanical arms is reduced.
2. According to the consistent tracking fixed time stability control method for the multi-single-link mechanical arm, the multi-single-link mechanical arm system can realize the convergence of synchronous errors in fixed time under different system initial states, the convergence time is irrelevant to the system initial state, and higher convergence speed and better tracking precision can be obtained by selecting proper parameters.
Drawings
FIG. 1 is a communication topology diagram of a control method according to an embodiment of the present invention;
FIG. 2 is a schematic diagram of the output of the reference signal and followers of the control method of the embodiment of the invention;
FIG. 3 is a schematic diagram of a synchronization error in an initial state 1 of a control method according to an embodiment of the present invention;
FIG. 4 is a schematic diagram of a synchronization error in initial state 2 of the control method according to the embodiment of the present invention;
FIG. 5 is a schematic diagram of a synchronization error in initial state 3 of the control method according to the embodiment of the present invention;
FIG. 6 is a schematic diagram of control signals of follower 1 according to the control method of the embodiment of the present invention;
FIG. 7 is a schematic diagram of control signals of follower 2 according to the control method of the embodiment of the present invention;
FIG. 8 is a schematic diagram of control signals of a follower 3 according to the control method of the embodiment of the present invention;
FIG. 9 is a schematic diagram of control signals of a follower 4 according to the control method of the embodiment of the present invention;
FIG. 10 is a schematic diagram of trigger event time intervals of follower 1 and follower 2 in the control method of the embodiment of the present invention;
fig. 11 is a schematic diagram of trigger event time intervals of the follower 3 and the follower 4 in the control method according to the embodiment of the present invention.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
Examples
Referring to fig. 1-11, the consistent tracking fixed time stability control method for a multi-single link mechanical arm according to the present invention first models an input dead zone, and establishes a set of single link mechanical arm system models with input dead zone constraints. Then, based on a back-stepping technology, a neural network self-adaptive control technology and a fixed time stability control theory, a new fixed time stability event trigger type controller is designed, and finally, the method is simulated. Simulation results show that each mechanical arm (follower) can well track the set reference signal, the control precision is good, and communication resources are effectively saved.
The method specifically comprises the following steps:
s01, modeling the single-link mechanical arm to obtain a state equation;
s02, defining the consistency tracking error of the ith single-link mechanical arm and designing a first virtual control law alpha i,1 ；
S03, designing a relative threshold event trigger mechanism;
s04, designing a second virtual law alpha i,2 And law of adaptation
Figure BDA0003648687500000081
And
Figure BDA0003648687500000082
and estimating unknown system parameters on line.
And S05, carrying out simulation experiments on the algorithm based on the Matlab experiment platform.
The further detailed steps are as follows:
s01, without loss of generality, the mathematical model of the ith single link arm in the follower is set as follows:
Figure BDA0003648687500000083
wherein t represents time, q i (t) is the joint angle (c),
Figure BDA0003648687500000084
is the angular velocity of the joint or joints,
Figure BDA0003648687500000085
is angular acceleration of the joint, J i Is moment of inertia, B i Is the coefficient of friction damping, m i Is the connecting rod mass, g is the gravitational acceleration,/ i Is the length of the connecting rod, u i (t) is an input torque.
Due to the structural characteristics of the mechanical structure, the mechanical arm system often has an input dead zone, which has a great influence on the system performance. The mathematical model for setting the input dead zone is as follows:
D i (u i )＝h i u i +g i (2)
wherein ,ui Represents a control input;
Figure BDA0003648687500000091
h i,1 、h i,2 、g i,1 and gi,2 Are all constant, and h i,1 ＞0、h i,2 ＞0、g i,1＜0 and gi,2 ＞0。
Coordinate transformation of (1) it can result in:
Figure BDA0003648687500000092
wherein ,xi,1 ＝q i (t)，
Figure BDA0003648687500000093
Each represents the system state of the i (i ═ 1, 2.., N) th follower;
Figure BDA0003648687500000094
denotes x i,1 A derivative of (a);
Figure BDA0003648687500000095
denotes x i.2 A derivative of (a); y is i A system output representing the i (i ═ 1, 2.., N) th follower;
Figure BDA0003648687500000096
indicating a system uncertainty portion.
In order to facilitate the description of the communication relationship between the mechanical arms in the topological diagram of fig. 2, relevant knowledge of algebraic graph theory needs to be introduced. Drawing (A)
Figure BDA00036486875000000920
Representing a directed communication topological graph of a multi-mechanical arm system, wherein each node in the graph corresponds to one mechanical arm, and omega is { omega ═ omega { (omega) } 1 ,Ω 2 ,...,Ω N Denotes the set of N nodes, the set of edges between nodes is
Figure BDA0003648687500000097
The edge from node i to node j is defined as an ordered pair
Figure BDA0003648687500000098
Indicating that the mechanical arm i can receive the information of the mechanical arm j, and calling the node i to be adjacent to the node j to define
Figure BDA0003648687500000099
Is the set of adjacent edges of agent i. A ═ a i,j ]∈R N×N Represents a adjacency matrix if
Figure BDA00036486875000000910
Then a i,j Is greater than 0; otherwise a i,j 0. The degree of entry of the node i is
Figure BDA00036486875000000911
Definition of
Figure BDA00036486875000000912
Figure BDA00036486875000000913
Is an in-degree diagonal matrix, then
Figure BDA00036486875000000914
Is a laplacian matrix.
To deal with unknown non-linear functions in the system, radial basis function neural networks are employed. In tight set omega ∈ R n Any continuous function defined above
Figure BDA00036486875000000915
Can be approximated by a neural network and can be expressed as:
Figure BDA00036486875000000916
wherein ,
Figure BDA00036486875000000917
is an ideal unknown weight vector, and
Figure BDA00036486875000000918
q is a positive integer;
Figure BDA00036486875000000919
represent
Figure BDA0003648687500000101
Transposing; Φ (X) is a vector of basis functions, and Φ (X) ═ Φ 1 (X),Φ 2 (X),...,Φ q (X)] T (ii) a q is the node number of the neural network, and q is more than 1; x represents an input vector, and X ═ X 1 ,x 2 ,...,x q ] T ∈R q (ii) a Mu (X) is an approximation error and satisfies
Figure BDA0003648687500000102
Figure BDA0003648687500000103
and Φi The definition of (X) is as follows:
Figure BDA0003648687500000104
Figure BDA0003648687500000105
wherein ,Wi ＝[W 1 ,W 2 ,...,W q ] T ∈R q Representing a weight vector;
Figure BDA0003648687500000106
and
Figure BDA00036486875000001017
are respectively a Gaussian function phi i The center and width of (X).
S02, first, defining the consistency tracking error of the ith mechanical arm by using the knowledge of graph theory:
Figure BDA0003648687500000107
wherein ,zi,1 For synchronization error, z i,2 Is a virtual control error, α i,1 Is the virtual control law, y d To be lost for the leaderAnd (6) outputting a signal. b i (i ═ 1, 2.., N) denotes the information transmission coefficient from the leader to the i-th follower. b i For positive numbers, if there is a transmission of information between the leader and the ith follower, b i Is greater than 0. Otherwise b i 0. As shown in the formula (7), the synchronization error z i,1 Weighted parameter a ij and bi The influence of (c). Therefore, a directed graph
Figure BDA0003648687500000108
Structural influence of (a) z i,1 . Furthermore, according to d i and bi Definition of (d) i +b i Is strictly positive. Further, the step S02 specifically includes:
s021, processing unknown combined parts by adopting radial basis function neural network
Figure BDA0003648687500000109
Introducing an unknown positive parameter
Figure BDA00036486875000001010
Wherein | | · | | represents a two-norm;
Figure BDA00036486875000001011
denotes y d The first derivative of (a); x i,1 Represents an input vector, an
Figure BDA00036486875000001012
x j,1 and xj,2 Represents the system state of the jth (j ═ 1, 2.., N) follower; and the parameter theta i,1 Can pass through
Figure BDA00036486875000001013
Estimate, i.e.
Figure BDA00036486875000001014
Is a parameter theta i,1 Then the final estimation error can be defined as
Figure BDA00036486875000001015
Thus, approximation processing
Figure BDA00036486875000001016
The expression of (a) is as follows:
Figure BDA0003648687500000111
wherein ,
Figure BDA0003648687500000112
is an ideal unknown weight vector, and
Figure BDA0003648687500000113
Figure BDA0003648687500000114
to represent
Figure BDA0003648687500000115
Transposing; phi (X) is a vector of basis functions, and phi i,1 (X i,1 )＝[Φ 1,1 (X i,1 ),Φ 2,1 (X i,1 ),...,Φ q,1 (X i,1 )] T (ii) a q is the node number of the neural network, and q is more than 1; mu.s i,1 (X i,1 ) Represents an approximation error, satisfies
Figure BDA0003648687500000116
And is
Figure BDA0003648687500000117
S022, first virtual control law α i,1 The design is as follows:
Figure BDA0003648687500000118
Figure BDA0003648687500000119
wherein ,ci,1 、r i,1 、a i,1 and εi,1 Are all constant, and c i,1 ＞0，r i,1 ＞0，a i,1 ＞0，ε i,1 ＞0。
S03, in order to save communication resources, a relative threshold event trigger mechanism is designed. Control signal omega i (t) is defined as follows:
Figure BDA00036486875000001110
only when the pre-designed trigger condition | e i (t)|≥ρ i |u i (t)|+ο i When true, controls the input signal u i (t) is updated, the expression is as follows:
Figure BDA00036486875000001111
wherein δi 、ο i 、ρ i 、
Figure BDA00036486875000001112
Are all positive design parameters and satisfy delta i ＞0，0＜ρ i ＜1，ο＞0，
Figure BDA00036486875000001113
k is an integer. inf {. } represents an infimum bound; t is t i,k The kth trigger time of the ith agent; t is t i,k+1 The (k + 1) th trigger time of the ith agent; e.g. of the type i (t) denotes measurement error, and e i (t)＝ω i (t)-u i (t)。
Because the system model has an uncertain part, the uncertain part of the second-order nonlinear system model is approximated through a neural network; according to the virtual control error z i,2 Design of virtual control law α i,2 And determining adaptive parameters. Further, the step S04 specifically includes:
s041, determining by using radial basis function neural networkUncertain combination parts in fixed system model
Figure BDA0003648687500000121
Figure BDA0003648687500000122
Denotes y d The second derivative of (a); x i,2 Represents an input vector, an
Figure BDA0003648687500000123
Introducing an unknown positive parameter
Figure BDA0003648687500000124
Wherein | · | | represents a two-norm; and the parameter theta i,2 Can pass through
Figure BDA0003648687500000125
Estimate, i.e.
Figure BDA0003648687500000126
Is a parameter theta i,2 Then the final estimation error can be defined as
Figure BDA0003648687500000127
Thus, approximation processing
Figure BDA0003648687500000128
The expression of (a) is as follows:
Figure BDA0003648687500000129
wherein ,
Figure BDA00036486875000001210
is an ideal unknown weight vector, and
Figure BDA00036486875000001211
Figure BDA00036486875000001212
to represent
Figure BDA00036486875000001213
Transposing;
Figure BDA00036486875000001214
is a vector of basis functions, and phi i,2 (X i,2 )＝[Φ 1,2 (X i,2 ),Φ 2,2 (X i,2 ),...,Φ q,2 (X i,2 )] T (ii) a q is the number of nodes of the neural network, and q is more than 1; mu.s i,2 (X i,2 ) Represents an approximation error, satisfies
Figure BDA00036486875000001215
And is
Figure BDA00036486875000001216
S042 according to the second error variable z i,2 In the process, the unknown uncertain part is obtained by solving the radial basis function neural network, and a second virtual control law alpha is designed by utilizing a backstepping design method and the Lyapunov function i,2 While generating adaptive parameters
Figure BDA00036486875000001217
And
Figure BDA00036486875000001218
in which a virtual control law alpha is established i,2 、
Figure BDA00036486875000001219
Law of control
Figure BDA00036486875000001220
And adaptive parameters
Figure BDA00036486875000001221
Figure BDA00036486875000001222
Satisfies the following calculation formula:
Figure BDA00036486875000001223
Figure BDA00036486875000001224
Figure BDA00036486875000001225
Figure BDA0003648687500000131
Figure BDA0003648687500000132
wherein
Figure BDA0003648687500000133
Represents phi i Transpose of (J) i 、ε i,2 、c i,2 、r i,2 、a i,2 、ξ i 、σ i 、l i,2 、η i And ζ i Are all positive design parameters. H i ＝[1/h i ,-g i /h i ] T ，H i T Represents H i Transpose of (Q) i ＝[α i,2 ,1] T ，α i,2 Representing a second virtual control law; since in practical cases, H i Are difficult to obtain and design
Figure BDA0003648687500000134
To estimate H i Defining an estimation error
Figure BDA0003648687500000135
And is
Figure BDA0003648687500000136
wherein
Figure BDA0003648687500000137
And
Figure BDA0003648687500000138
are respectively estimated as
Figure BDA0003648687500000139
And
Figure BDA00036486875000001310
Figure BDA00036486875000001311
multiplication by multiplication
Figure BDA00036486875000001312
And S05, simulating the algorithm based on a Matlab experiment platform in order to verify the effectiveness of the method. FIG. 1 depicts a communication topology with one virtual leader and four followers. In this figure, arms 1-4 and arm d represent four followers and a virtual leader, respectively. Without loss of generality, assume that the output signal of the virtual leader (reference signal) is y d Sin (2 t). According to the basic principle of graph theory, an adjacency matrix a and a laplacian matrix L can be obtained:
Figure BDA00036486875000001313
to deal with the uncertain part, a radial basis function neural network is used, the Gaussian function of which is phi i (X i ) The expression is as follows:
Figure BDA00036486875000001314
the control objective of the simulation experiment is to make the joint angular velocity
Figure BDA00036486875000001315
Tracking the set track signal y d Sin (2 t). The parameters of the relevant system are as follows: moment of inertia J i 0.8, damping coefficient B i ＝1，m i gl i 10, i is 1,2,3, 4. Unknown part of the system
Figure BDA00036486875000001316
The initial state of the system is shown in table 1.
The set values of the relevant parameters are as follows:
Figure BDA00036486875000001317
c 1,1 ＝r 1,1 ＝14，c 1,2 ＝r 1,2 ＝3，c 2,1 ＝r 2,1 ＝25，c 2,2 ＝r 2,2 ＝2，c 3,1 ＝r 3,1 ＝25，c 3,2 ＝r 3,2 ＝1，c 4,1 ＝r 4,1 ＝26，c 4,2 ＝r 4,2 ＝1，ε i,1 ＝ε i,2 ＝0.01，l i,1 ＝a i,1 ＝1，ξ i ＝σ i ＝0.2，η i ＝ζ i ＝0.01，K i ＝[1,0；0,1]，δ i ＝0.1，ρ i ＝0.01，ο i ＝3，
Figure BDA0003648687500000141
and i is 1. The initial estimates are:
Figure BDA0003648687500000142
and i is 1. The simulated sampling period is 0.01 s.
TABLE 1 three initial states of the system
Initial state x 1,1 (0) x 1,2 (0) x 2,1 (0) x 2,2 (0) x 3,1 (0) x 3,2 (0) x 4,1 (0) x 4,2 (0)
Initial state 1 0.1 0 0.1 0 0 0 -0.1 0
Initial state 2 0.1 0 0.1 0 -0.1 0 0.1 0
Initial State 3 0.1 0 0.1 0 -0.1 0 -0.1 0
TABLE 2 number of triggers for each follower
Figure BDA0003648687500000143
The simulation results are shown in fig. 2-9. As can be seen from fig. 2-9, all signals are bounded. FIG. 2 shows the output signals of the leader and follower, indicating good consistency tracking performance. The synchronization error curve of state 1 is shown in fig. 3, and it can be obtained that the synchronization error enters a 5% error band after about 0.05s, which indicates that the convergence rate is fast. In addition, the synchronization errors for the other two initial states (state 2 and state 3 in Table 1) are shown in FIGS. 4-5. Simulation results show that the convergence time of the synchronization errors in the three initial states is about 0.05s, namely the convergence time is not influenced by the initial state of the system. Control input signals for four followers As shown in FIGS. 6-9, the event-triggered control input w (t) is made continuously smooth, while the control input u (t) and the input dead band D i (u i ) Is serrated. Event trigger statistics are shown in table 2, and the event trigger time intervals for each follower are shown in fig. 10-11, respectively. As can be seen from table 2, the total number of triggers based on the conventional Time Trigger Mechanism (TTM) and Event Trigger Mechanism (ETM) is 4000 and 1033 times, respectively. Thus, the event trigger rate is 25.8%, which means a saving of 74.2% of communication resources. From FIGS. 10-11, it can be seen that the minimum trigger interval for all followers is 0.01s, and obviously, noneThere is a Zeno phenomenon.
According to the invention, a new relative threshold event trigger mechanism is designed in the controller, so that on-line dynamic compensation of an input dead zone can be realized on the premise of ensuring the control precision, and the update frequency of input signals of each mechanical arm is reduced; the multi-single-link mechanical arm system can realize the convergence of synchronous errors in fixed time under different system initial states, the convergence time is irrelevant to the system initial state, and higher convergence speed and better tracking accuracy are obtained by selecting proper parameters.
It is noted that, herein, relational terms such as first and second, and the like may be used solely to distinguish one entity or action from another entity or action without necessarily requiring or implying any actual such relationship or order between such entities or actions. Also, the terms "comprises," "comprising," or any other variation thereof, are intended to cover a non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements does not include only those elements but may include other elements not expressly listed or inherent to such process, method, article, or apparatus. The term "comprising", without further limitation, means that the element so defined is not excluded from the group consisting of additional identical elements in the process, method, article, or apparatus that comprises the element.

Claims (6)

1. A consistent tracking fixed time stability control method of a multi-single-link mechanical arm is characterized by comprising the following steps:
s01, modeling the single-link mechanical arm to obtain a state equation;
s02, defining the consistency tracking error of the ith single-link mechanical arm and designing a first virtual control law alpha i,1 ；
S03, designing a relative threshold event trigger mechanism;
s04, designing a second virtual law alpha i,2 And law of adaptation
Figure FDA0003648687490000011
And
Figure FDA0003648687490000012
estimating unknown system parameters on line;
and S05, carrying out simulation experiments on the algorithm based on the Matlab experiment platform.
2. The method for controlling the consistent tracking and fixed time stability of a multi-single link robot arm according to claim 1, wherein the step S01 is as follows:
without loss of generality, the mathematical model of the ith single link arm in the follower is set up as follows:
Figure FDA0003648687490000013
wherein t represents time, q i (t) is the joint angle (c),
Figure FDA0003648687490000014
is the angular velocity of the joint or joints,
Figure FDA0003648687490000015
is angular acceleration of the joint, J i Is moment of inertia, B i Is the coefficient of friction damping, m i Is the connecting rod mass, g is the gravitational acceleration,/ i Is the length of the connecting rod, u i (t) is input torque;
due to the structural characteristics of the mechanical structure, the mechanical arm system often has an input dead zone, which has great influence on the system performance; the mathematical model for setting the input dead zone is as follows:
D i (u i )＝h i u i +g i (2)
wherein ,ui Represents a control input;
Figure FDA0003648687490000016
h i,1 、h i,2 、g i,1 and gi,2 Are all constant, and h i,1 ＞0、h i,2 ＞0、g i,1＜0 and gi,2 ＞0；
Coordinate transformation of equation (1) can result in:
Figure FDA0003648687490000021
wherein ,xi,1 ＝q i (t)，
Figure FDA0003648687490000022
Each represents the system state of the i (i ═ 1, 2.., N) th follower;
Figure FDA0003648687490000023
denotes x i,1 A derivative of (a);
Figure FDA0003648687490000024
denotes x i.2 A derivative of (a); y is i A system output representing the i (i ═ 1, 2.., N) th follower;
Figure FDA0003648687490000025
representing a system uncertainty portion;
in order to facilitate the description of the communication relationship between the mechanical arms in the topological graph, relevant knowledge of algebraic graph theory needs to be introduced. Drawing (A)
Figure FDA0003648687490000026
Representing a directed communication topological graph of a multi-mechanical arm system, wherein each node in the graph corresponds to one mechanical arm, and omega is { omega ═ omega { (omega) } 1 ,Ω 2 ,...,Ω N Denotes the set of N nodes, the set of edges between nodes is
Figure FDA0003648687490000027
Edge from node i to node jIs defined as an ordered pair
Figure FDA0003648687490000028
Indicating that the mechanical arm i can receive the information of the mechanical arm j, and calling the node i to be adjacent to the node j to define
Figure FDA0003648687490000029
Is the set of adjacent edges of agent i. A ═ a i,j ]∈R N×N Represents a adjacency matrix if
Figure FDA00036486874900000210
Then a i,j Is greater than 0; otherwise a i,j 0. The degree of entry of the node i is
Figure FDA00036486874900000211
Definition of
Figure FDA00036486874900000212
Figure FDA00036486874900000213
Is an in-degree diagonal matrix, then
Figure FDA00036486874900000214
Is a laplace matrix;
in order to process unknown nonlinear functions in the system, a radial basis function neural network is adopted; in tight set omega ∈ R n Any continuous function defined above
Figure FDA00036486874900000215
Can be approximated by a neural network and can be expressed as:
Figure FDA00036486874900000216
wherein ,
Figure FDA00036486874900000217
is an ideal unknown weight vector, and
Figure FDA00036486874900000218
q is a positive integer;
Figure FDA00036486874900000219
to represent
Figure FDA00036486874900000220
Transposing; Φ (X) is a vector of basis functions, and Φ (X) ═ Φ 1 (X),Φ 2 (X),...,Φ q (X)] T (ii) a q is the node number of the neural network, and q is more than 1; x represents an input vector, and X ═ X 1 ,x 2 ,...,x q ] T ∈R q (ii) a Mu (X) is an approximation error and satisfies
Figure FDA00036486874900000221
Figure FDA00036486874900000222
and Φi The definition of (X) is as follows:
Figure FDA00036486874900000223
Figure FDA0003648687490000031
wherein ,Wi ＝[W 1 ,W 2 ,...,W q ] T ∈R q Representing a weight vector;
Figure FDA0003648687490000032
and
Figure FDA0003648687490000033
are respectively a Gaussian function phi i The center and width of (X).
3. The method for controlling consistent tracking and fixed time stability of a multi-single link robot arm according to claim 1, wherein: the step S02 is as follows:
first, the consistency tracking error of the ith robot arm is defined by the knowledge of graph theory:
Figure FDA0003648687490000034
wherein ,zi,1 For synchronization error, z i,2 Is a virtual control error, α i,1 Is the virtual control law, y d Outputting a signal for the leader. b i (i ═ 1, 2.., N) denotes the information transmission coefficient from the leader to the i-th follower. b i For positive numbers, if there is a transmission of information between the leader and the ith follower, b i Is greater than 0. Otherwise b i 0. As shown in the formula (7), the synchronization error z i,1 Weighted parameter a ij and bi The influence of (c). Therefore, a directed graph
Figure FDA0003648687490000035
Structural influence of (a) z i,1 . Furthermore, according to d i and bi Definition of (d) i +b i Is strictly positive.
4. The method for controlling consistent tracking and fixed time stabilization of a multi-single link robot arm according to claim 3, wherein: the step S02 specifically includes:
s021, processing unknown combined parts by adopting radial basis function neural network
Figure FDA0003648687490000036
Introducing an unknown positive parameter
Figure FDA0003648687490000037
Wherein | · | | represents a two-norm;
Figure FDA0003648687490000038
denotes y d The first derivative of (a); x i,1 Represents an input vector, an
Figure FDA0003648687490000039
x j,1 and xj,2 Represents the system state of the jth (j ═ 1, 2.., N) follower; and the parameter theta i,1 Can pass through
Figure FDA00036486874900000310
Estimate, i.e.
Figure FDA00036486874900000311
Is a parameter theta i,1 Then the final estimation error can be defined as
Figure FDA00036486874900000312
Thus, approximation processing
Figure FDA00036486874900000313
The expression of (a) is as follows:
Figure FDA0003648687490000041
wherein ,
Figure FDA0003648687490000042
is an ideal unknown weight vector, and
Figure FDA0003648687490000043
Figure FDA0003648687490000044
to represent
Figure FDA0003648687490000045
Transposing; phi (X) is a vector of basis functions, and phi i,1 (X i,1 )＝[Φ 1,1 (X i,1 ),Φ 2,1 (X i,1 ),...,Φ q,1 (X i,1 )] T (ii) a q is the number of nodes of the neural network, and q is more than 1; mu.s i,1 (X i,1 ) Represents an approximation error, satisfies
Figure FDA0003648687490000046
And is
Figure FDA0003648687490000047
S022, first virtual control law α i,1 The design is as follows:
Figure FDA0003648687490000048
Figure FDA0003648687490000049
wherein ,ci,1 、r i,1 、a i,1 and εi,1 Are all constant, and c i,1 ＞0，r i,1 ＞0，a i,1 ＞0，ε i,1 ＞0。
5. The method for controlling the consistent tracking and fixed time stability of a multi-single link robot arm according to claim 1, wherein the step S03 is as follows:
in order to save communication resources, a relative threshold event trigger mechanism is designed, and a signal omega is controlled i (t) is defined as follows:
Figure FDA00036486874900000410
only when the pre-designed trigger condition | e i (t)|≥ρ i |u i (t)|+ο i When true, controls the input signal u i (t) is updated, the expression is as follows:
Figure FDA00036486874900000411
wherein δi 、ο i 、ρ i 、
Figure FDA00036486874900000412
Are all positive design parameters and satisfy delta i ＞0，0＜ρ i ＜1，ο＞0，
Figure FDA00036486874900000413
k is an integer. inf {. } represents an infimum bound; t is t i,k The kth trigger time of the ith agent; t is t i,k+1 The (k + 1) th trigger time of the ith agent; e.g. of the type i (t) denotes measurement error, and e i (t)＝ω i (t)-u i (t)。
6. The method for controlling the consistent tracking and fixed time stability of the multi-single-link mechanical arm according to claim 1, wherein the uncertain part of the second-order nonlinear system model is approximated by a neural network due to the existence of the uncertain part in the system model; designing a virtual control law according to the virtual control error, and determining an adaptive parameter, where step S04 specifically includes:
determining uncertain combination parts in system model by radial basis function neural network
Figure FDA0003648687490000051
Figure FDA0003648687490000052
Denotes y d The second derivative of (a); x i,2 Represents an input vector, an
Figure FDA0003648687490000053
Introducing an unknown positive parameter
Figure FDA0003648687490000054
Wherein | · | | represents a two-norm; and the parameter theta i,2 Can pass through
Figure FDA0003648687490000055
Estimate, i.e.
Figure FDA0003648687490000056
Is a parameter theta i,2 Then the final estimation error can be defined as
Figure FDA0003648687490000057
Thus, approximation processing
Figure FDA0003648687490000058
The expression of (a) is as follows:
Figure FDA0003648687490000059
wherein ,
Figure FDA00036486874900000510
is an ideal unknown weight vector, and
Figure FDA00036486874900000511
Figure FDA00036486874900000512
to represent
Figure FDA00036486874900000513
Transposing;
Figure FDA00036486874900000514
is a vector of basis functions, and phi i,2 (X i,2 )＝[Φ 1,2 (X i,2 ),Φ 2,2 (X i,2 ),...,Φ q,2 (X i,2 )] T (ii) a q is the node number of the neural network, and q is more than 1; mu.s i,2 (X i,2 ) Represents an approximation error, satisfies
Figure FDA00036486874900000515
And is
Figure FDA00036486874900000516
S042 according to the second error variable z i,2 In the process, the unknown uncertain part is obtained by solving the radial basis function neural network, and a second virtual control law alpha is designed by utilizing a backstepping design method and the Lyapunov function i2 While generating adaptive parameters
Figure FDA00036486874900000517
And
Figure FDA00036486874900000518
in which a virtual control law alpha is established i,2 、
Figure FDA00036486874900000519
Law of control
Figure FDA00036486874900000520
And adaptive parameters
Figure FDA00036486874900000521
Satisfies the following calculation formula:
Figure FDA00036486874900000522
Figure FDA00036486874900000523
Figure FDA00036486874900000524
Figure FDA00036486874900000525
Figure FDA0003648687490000061
wherein
Figure FDA0003648687490000062
Represents phi i Transpose of (J) i 、ε i,2 、c i,2 、r i,2 、a i,2 、ξ i 、σ i 、l i,2 、η i And ζ i Are all positive design parameters. H i ＝[1/h i ,-g i /h i ] T ，
Figure FDA0003648687490000063
Represents H i Transpose of (Q) i ＝[α i,2 ,1] T ，α i,2 Representing a second virtual control law; since in practical cases, H i Are difficult to obtain and design
Figure FDA0003648687490000064
To estimate H i Defining an estimation error
Figure FDA0003648687490000065
And is
Figure FDA0003648687490000066
wherein
Figure FDA0003648687490000067
And
Figure FDA0003648687490000068
are respectively estimated as
Figure FDA0003648687490000069
And
Figure FDA00036486874900000610
Figure FDA00036486874900000611
multiplication by multiplication
Figure FDA00036486874900000612
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CN116679573B (en) * 2023-08-04 2023-10-20 北京全路通信信号研究设计院集团有限公司 Consistency tracking control method and device, electronic equipment and storage medium

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