A finite-time control method for consistency tracking of a multi-link manipulator

Abstract

The present invention discloses a finite-time control method for consistent tracking of a multi-link robotic arm. The method comprises the following steps: first, modeling actuator failures, and then establishing an uncertain multi-link robotic arm system model with actuator failures and full-state constraints. Next, a finite-time stable event-triggered control method is designed based on barrier Lyapunov functions, backstepping design, and neural network adaptive control technology. Finally, the proposed control method is simulated. The simulation results show that when an actuator failure occurs, each robotic arm can effectively track the leader's trajectory, with minimal tracking error and stability within the specified time, while also effectively saving communication resources.

Description

Technical Field

[0001] The present invention belongs to the field of robotic arm control, and in particular relates to a consistency tracking finite time control method for a multi-link robotic arm. Background Art

[0002] In recent years, with the widespread application of multi-agent systems, the consistency problem of multi-link manipulator systems has attracted considerable attention. Multi-link manipulator systems enable multiple manipulators to work collaboratively, completing tasks that a single manipulator cannot accomplish. Each manipulator is connected via a topological network, enabling intercommunication between the manipulators. In practical applications, the position and velocity of multi-link systems are subject to certain constraints due to physical limitations or the need for safe operation. Failure to adhere to these constraints in control design can lead to system performance degradation and potential safety issues. Therefore, the study of constrained control is crucial. Furthermore, the actuators of single-link manipulators are connected to devices and are subject to various external conditions, making it crucial to ensure the effectiveness of control methods under various circumstances (such as actuator failures). Furthermore, traditional sampling methods for communication between manipulators often consume significant communication resources, causing network congestion. Therefore, an event-triggered mechanism is needed to conserve communication resources. Finally, achieving finite-time stability is also a key consideration in practical manipulator control systems. In summary, for an uncertain multi-link robotic arm system with full-state constraints and actuator failures, a neural network finite-time event-triggered control method is studied to achieve consistency control, which is of great significance for practical engineering applications.

[0003] Disadvantages of existing technology:

[0004] When studying the consistency of uncertain multi-manipulator systems, the positions and velocities of the manipulators are subject to constraints. Existing methods for achieving system consistency under these constraints assume that the actuators are operating normally. However, actuators in real systems are susceptible to environmental influences. Therefore, actuator failures must be considered and compensated for when designing control systems.

[0005] On the other hand, many existing control system approaches have infinite convergence times, which cannot meet the time constraints required in practical production. Furthermore, existing methods for achieving finite-time convergence require continuous controller updates and ongoing communication between the manipulators. However, in practical applications, communication resources, computational resources, and energy availability between the manipulators are all limited. Therefore, balancing control performance and conserving communication resources is crucial. Summary of the Invention

[0006] The object of the present invention is to provide a finite-time control method for consistency tracking of a multi-link robotic arm to solve the above-mentioned problem.

[0007] The present invention provides the following technical solutions:

[0008] A finite-time control method for consistency tracking of a multi-link robotic arm comprises the following steps:

[0009] S01: Establish a multi-link robot arm model and an actuator fault model, and combine the two to obtain a dynamic model of the multi-link robot arm system with actuator faults;

[0010] S02: Design the consistent tracking error system of the multi-link manipulator system and derive the first virtual control law υ i,1 and adaptive law

[0011] S03. Design a relative threshold event trigger mechanism to reduce the signal update frequency;

[0012] S04, through the adaptive law and The second virtual control law υ is derived i,2 ,υ i,2 Used to online approximate unknown system parameters to control multi-link robotic arms.

[0013] Preferably, in S01, the dynamic model of the multi-link robotic arm system is described by the following differential equation:

[0014]

[0015] Among them, i∈{1, 2, 3, 4}, represents i robotic arms; x i，1 and x i，2 are the displacement and velocity of the single-link manipulator, and x i，1 and x i，2 The derivative of u i and y i Represent the input and output of the i-th manipulator, J is the inertia of the single-link manipulator, M is the mass of the single-link manipulator, B is the viscous friction coefficient, G is the acceleration of gravity, L is the length between the joint and the center of mass, and function F1 represents the input disturbance; and x i，1 <1.52, x i，2 <3.5.

[0016] More preferably, in S02, the consistency tracking error system of the multi-link manipulator system is defined by the following formula:

[0017]

[0018] Where z i,1 ,z i,2 is the error variable, υ i,1 is the first virtual control law, is the transfer function, E i is the synchronization error of the i-th robot arm, Among them, y d is the trajectory of the virtual leader, y k represents the output of the k-th robotic arm, b i is the weight of the edge from the i-th robot to the leader. When the robot i receives the information from the leader, b i > 0, otherwise, b i =0;a ik Represents the weight of the edge from the i-th robotic arm to the k-th robotic arm. When the information of the k-th robotic arm can be transmitted to the i-th robotic arm, a ik > 0, otherwise a ik =0.

[0019] Better, the first virtual control law υ i,1 and adaptive control law as follows:

[0020]

[0021] where μ i,1 and σ are both positive design parameters, where 0<σ<1, is a greater than |z i,1 | a positive number, For unknown weight vector It can be estimated using the adaptive law, assuming It's Q i,1 The estimated value of

[0022]

[0023] in φ i,1 , q i,1 and δ i,1 are all positive design parameters, z i,1 represents the estimation error, ω i,1,l represents radial basis function.

[0024] More preferably, in S03, in order to improve the utilization rate of communication resources and realize the conservation of communication resources, a relative threshold event triggering mechanism is designed. The mathematical model expression of the event triggering mechanism is as follows:

[0025]

[0026] Among them, ω i,j (t) represents the event-triggered control input, h i,j represents the unknown parameters in the system, γ i,j (x i ) represents the known nonlinear function in the system, ε, ε1, and λ are both positive design parameters, λ>0,0<ε<1, u dj Indicates the control signal, Indicates the difference between the control signal and the actuator input; only when the pre-designed trigger condition When satisfied, the control input signal Will be updated, then there is t k+1 is the triggering moment, t k Indicates the last trigger moment, otherwise R represents the set of real numbers.

[0027] More preferably, in S04, the second virtual control law υ i,2 , adaptive control law and Satisfies the following calculation formula:

[0028]

[0029] in, q i,2 , and μ i,2 are all positive design parameters, ω i,2,l represents the radial basis function, z i,2 represents the error variable, is a greater than |z i,2 | a positive number, It's Q i,2 estimated value of;

[0030]

[0031]

[0032] Where 0<σ<1, φ i,2 , χ i,j , δ i,2 and are all positive design parameters, is ρ i,j estimated value.

[0033] Compared with the existing methods, the present invention has the following advantages:

[0034] 1. For uncertain multi-link robotic arm systems, the present invention simultaneously considers full-state constraints and actuator failures, and proposes a neural network adaptive consistency control method based on obstacle Lyapunov function and radial basis function neural network. The follower robotic arm can converge to the specified trajectory of the leader within a finite time, with a faster convergence speed and better tracking performance.

[0035] 2. In order to save the communication pressure between each robotic arm, the present invention designs an event-triggered control mechanism to achieve intermittent communication and intermittent controller updates between adjacent robotic arms, effectively saving system communication resources. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] The present invention is further described with reference to the accompanying drawings. However, the embodiments in the accompanying drawings do not constitute any limitation to the present invention. A person skilled in the art can obtain other drawings based on the following drawings without creative effort.

[0037] Figure 1 Flowchart of a consistent tracking finite time control method for a multi-link robotic arm according to an embodiment of the present invention;

[0038] Figure 2 is a communication topology diagram provided by an embodiment of the present invention;

[0039] Figure 3 The reference output y in the first case provided by the embodiment of the present invention is d and the system output y i Trajectory diagram;

[0040] Figure 4 The synchronization error E in the first case provided by the embodiment of the present invention is i Trajectory diagram;

[0041] Figure 5 The system state x in the first case provided by the embodiment of the present invention is 1，2 Trajectory diagram;

[0042] Figure 6 The system state x in the first case provided by the embodiment of the present invention is 2，2 Trajectory diagram;

[0043] Figure 7 The system state x in the first case provided by the embodiment of the present invention is 3，2 Trajectory diagram;

[0044] Figure 8 The system state x in the first case provided by the embodiment of the present invention is 4，2 Trajectory diagram;

[0045] Figure 9 This is the event-triggered control input ω in the first case provided by the embodiment of the present invention. 1，1 (t) and system input u 1，1 (t) trajectory diagram;

[0046] Figure 10 This is the event-triggered control input ω in the first case provided by the embodiment of the present invention. 2，1 (t) and system input u 2，1 (t) trajectory diagram;

[0047] Figure 11 This is the event-triggered control input ω in the first case provided by the embodiment of the present invention. 3，1 (t) and system input u 3，1 (t) trajectory diagram;

[0048] Figure 12 This is the event-triggered control input ω in the first case provided by the embodiment of the present invention. 4，1 (t) and system input u 4，1 (t) trajectory diagram;

[0049] Figure 13 This is the event-triggered control input ω in the first case provided by the embodiment of the present invention. 1，2 (t) and system input u 1，2 (t) trajectory diagram;

[0050] Figure 14 This is the event-triggered control input ω in the first case provided by the embodiment of the present invention. 2，2 (t) and system input u 2，2 (t) trajectory diagram;

[0051] Figure 15 This is the event-triggered control input ω in the first case provided by the embodiment of the present invention. 3，2 (t) and system input u 3，2 (t) trajectory diagram;

[0052] Figure 16 This is the event-triggered control input ω in the first case provided by the embodiment of the present invention. 4，2 (t) and system input u 4，2 (t) trajectory diagram;

[0053] Figure 17 1 is a schematic diagram of the execution interval time of agent 1 in the first case provided by an embodiment of the present invention;

[0054] Figure 18 1 is a schematic diagram of the execution interval time of agent 2 in the first case provided by an embodiment of the present invention;

[0055] Figure 19 Schematic diagram of the execution interval time of agent 3 in the first case provided by an embodiment of the present invention;

[0056] Figure 20 1 is a schematic diagram of the execution interval time of the agent 4 in the first case provided by the embodiment of the present invention;

[0057] Figure 21 is the reference output y in the second case provided by the embodiment of the present invention d and the system output y i Trajectory diagram;

[0058] Figure 22 The synchronization error E in the second case provided by the embodiment of the present invention is i Trajectory diagram;

[0059] Figure 23 The system state x in the second case provided by the embodiment of the present invention is 1，2 Trajectory diagram;

[0060] Figure 24 The system state x in the second case provided by the embodiment of the present invention is 2，2 Trajectory diagram;

[0061] Figure 25 The system state x in the second case provided by the embodiment of the present invention is 3，2 Trajectory diagram;

[0062] Figure 26 The system state x in the second case provided by the embodiment of the present invention is 4，2 Trajectory diagram;

[0063] Figure 27 is the event-triggered control input ω in the second case provided by the embodiment of the present invention 1，1 (t) and system input u 1，1 (t) trajectory diagram;

[0064] Figure 28 is the event-triggered control input ω in the second case provided by the embodiment of the present invention 2，1 (t) and system input u 2，1 (t) trajectory diagram;

[0065] Figure 29 is the event-triggered control input ω in the second case provided by the embodiment of the present invention 3，1 (t) and system input u 3，1 (t) trajectory diagram;

[0066] Figure 30 is the event-triggered control input ω in the second case provided by the embodiment of the present invention 4，1(t) and system input u 4，1 (t) trajectory diagram;

[0067] Figure 31 is the event-triggered control input ω in the second case provided by the embodiment of the present invention 1，2 (t) and system input u 1，2 (t) trajectory diagram;

[0068] Figure 32 is the event-triggered control input ω in the second case provided by the embodiment of the present invention 2，2 (t) and system input u 2，2 (t) trajectory diagram;

[0069] Figure 33 is the event-triggered control input ω in the second case provided by the embodiment of the present invention 3，2 (t) and system input u 3，2 (t) trajectory diagram;

[0070] Figure 34 is the event-triggered control input ω in the second case provided by the embodiment of the present invention 4，2 (t) and system input u 4，2 (t) trajectory diagram;

[0071] Figure 35 1 is a schematic diagram of the execution interval time of agent 1 in the second case provided by an embodiment of the present invention;

[0072] Figure 36 1 is a schematic diagram of the execution interval time of agent 2 in the second case provided by an embodiment of the present invention;

[0073] Figure 37 1 is a schematic diagram of the execution interval time of the agent 3 in the second case provided by an embodiment of the present invention;

[0074] Figure 38 1 is a schematic diagram of the execution interval time of the agent 4 in the second case provided by an embodiment of the present invention; DETAILED DESCRIPTION

[0075] The following is a further detailed description of the consistency tracking finite time control method for a multi-link robotic arm with actuator failure in conjunction with specific embodiments. These embodiments are only for comparison and explanation purposes, and the present invention is not limited to these embodiments.

[0076] Example:

[0077] See attached Figure 1-38 The present invention provides a consistent tracking finite time control method for a multi-link robotic arm, comprising the following steps:

[0078] S01: Establish a multi-link robot arm model and an actuator fault model, and combine the two to obtain a dynamic model of the multi-link robot arm system with actuator faults;

[0079] S02: Design the consistent tracking error system of the multi-link manipulator system and derive the first virtual control law υ i,1 and adaptive law

[0080] S03. Design a relative threshold event trigger mechanism to reduce the signal update frequency;

[0081] S04, through the adaptive law and The second virtual control law υ is derived i,2 ,υ i,2 Used to online approximate unknown system parameters to control multi-link robotic arms.

[0082] Preferably, in S01, the dynamic model of the multi-link robotic arm system is described by the following differential equation:

[0083]

[0084] Among them, i∈{1, 2, 3, 4}, represents i robotic arms, i.e. i intelligent agents; x i，1 and x i，2 are the displacement and velocity of the single-link manipulator, and x i，1 and x i，2 The derivative of u i and y i Represent the input and output of the i-th manipulator, J is the inertia of the single-link manipulator, M is the mass of the single-link manipulator, B is the viscous friction coefficient, G is the acceleration of gravity, L is the length between the joint and the center of mass, and function F1 represents the input disturbance; and x i，1 <1.52, x i，2 <3.5.

[0085] More preferably, in S02, the consistency tracking error system of the multi-link manipulator system is defined by the following formula:

[0086]

[0087] Where z i,1 ,z i,2 is the error variable, υ i,1 is the first virtual control law, is the transfer function, E i is the synchronization error of the i-th robot arm, Among them, yd is the trajectory of the virtual leader, y k represents the output of the k-th robotic arm, b i is the weight of the edge from the i-th robot to the leader. When the robot i receives the information from the leader, b i > 0, otherwise, b i =0;a ik Represents the weight of the edge from the i-th robotic arm to the k-th robotic arm. When the information of the k-th robotic arm can be transmitted to the i-th robotic arm, a ik > 0, otherwise a ik =0.

[0088] Better, the first virtual control law υ i,1 and adaptive control law as follows:

[0089]

[0090] where μ i,1 and σ are both positive design parameters, where 0<σ<1, is a greater than |z i,1 | a positive number, For unknown weight vector It can be estimated using the adaptive law, assuming It's Q i,1 The estimated value of

[0091]

[0092] in φ i,1 , q i,1 and δ i,1 are all positive design parameters, z i,1 represents the estimation error, ω i,1,l represents radial basis function.

[0093] More preferably, in S03, in order to improve the utilization rate of communication resources and realize the conservation of communication resources, a relative threshold event triggering mechanism is designed. The mathematical model expression of the event triggering mechanism is as follows:

[0094]

[0095] Among them, ω i,j( t) represents the event trigger control input, h i,j represents the unknown parameters in the system, γ i,j (x i ) represents the known nonlinear function in the system, ε, ε1, and λ are both positive design parameters, λ>0,0<ε<1, u dj Indicates the control signal, Indicates the difference between the control signal and the actuator input; only when the pre-designed trigger condition When satisfied, the control input signal Will be updated, then there is t k+1 is the triggering moment, t k Indicates the last trigger moment, otherwise R represents the set of real numbers.

[0096] More preferably, in S04, the second virtual control law υ i,2 , adaptive control law and Satisfies the following calculation formula:

[0097]

[0098] in, q i,2 , and μ i,2 are all positive design parameters, ω i,2,l represents the radial basis function, z i,2 represents the error variable, is a greater than |z i,2 | a positive number, It's Q i,2 estimated value of;

[0099]

[0100]

[0101] Where 0<σ<1, φ i,2 , χ i,j , δ i,2 and are all positive design parameters, is ρ i,j estimated value.

[0102] In one embodiment, if Figure 1 As shown, S01: establish a multi-link robot arm model and an actuator fault model, and combine the two to obtain a multi-link robot arm system dynamic model with actuator fault;

[0103] The dynamic model of the multi-link manipulator system with uncertain parameters is described by the following differential equation:

[0104]

[0105] Assuming the system has 4 robotic arms, then i = 1,…,4, which represents the number of agents, i.e. the number of robotic arms; x i,1 and x i,2 Represents the system state, specifically the displacement and velocity of the single-link robotic arm, u i and y i represents the input and output of the i-th manipulator, J is the inertia of the single-link manipulator, M is the mass of the single-link manipulator, B is the viscous friction coefficient, G is the acceleration due to gravity, and L is the length between the joint and the center of mass. Function F1 represents the input disturbance. Since the displacement and velocity of the manipulator are constrained in actual engineering, the state of the system needs to satisfy |x i,1 |<1.52、|x i,2 |<3.5.

[0106] The actuator of the single-link robot arm is connected to the device and is affected by various external conditions. The actuator of the robot arm may fail. Assuming that each agent contains m actuators, the input u of the i-th robot arm can be obtained. i for:

[0107]

[0108] Where, j=1,…,m represents the number of actuators, h i,j represents the unknown parameters in the system, γ i,j (x i ) represents a known nonlinear function in the system, u i,j represents the input of the jth actuator of the i-th robotic arm. The fault mathematical model of the jth actuator of the i-th robotic arm is defined as follows:

[0109]

[0110] where c i,j ∈[0,1], represents the degree of failure of the actuator, θ i,j and t jF is an uncertain constant, t jF represents the time when the actuator fails, t represents time, θ i,j Indicates the output when the actuator fails, Indicates the output when the actuator is operating normally.

[0111] The actuator has three main states:

[0112] 1)c i,j =1: The actuator is in normal state during operation.

[0113] 2) 0<c i,j<1: The actuator has a partial fault during operation.

[0114] 3)c i,j =0: The actuator fails completely during operation. i,j =θ i,j .

[0115] Combining the multi-link robot system model and the actuator fault model, the expression of the multi-link robot system model with full state constraints and actuator faults is determined as follows:

[0116]

[0117] S02: Design the consistent tracking error system of the multi-link manipulator system and derive the first virtual control law υ i,1 and adaptive law

[0118] First, define the synchronization error of the i-th robotic arm as:

[0119]

[0120] Where y d is the trajectory of the virtual leader, that is, its output signal. b i is the weight of the edge from the i-th robot to the leader. When the robot i receives the information from the leader, b i > 0, otherwise, b i =0. a ik Represents the weight of the edge from the i-th robotic arm to the k-th robotic arm. When the information of the k-th robotic arm can be transmitted to the i-th robotic arm, a ik > 0, otherwise a ik =0.

[0121] Furthermore, the step S02 specifically includes:

[0122] S021. Transform the error system and define the following error transformation system:

[0123]

[0124] Where z i,1 ,z i,2 represents the error variable of the i-th robotic arm, υ i,1 is the virtual control law, is the conversion function. choose:

[0125]

[0126] Where π, Θ and Δ are positive design parameters, that is, π>0, Θ>0, Δ>0, t represents time, and the The main purpose is to obtain the mapping and amplification of the synchronization error signal.

[0127] S022. Determine the uncertain unknown combination parts in the system by using radial basis function neural network x k,1 and x k,2 Represents the system state of the (k=1,2,3,4)th robotic arm; represents y d The first-order derivative of can be approximated The expression is as follows:

[0128]

[0129] Where l = 1, 2, ..., 9 represents the number of neural nodes. represents the ideal weight vector, Yes i,1 The transposed matrix of

[0130]

[0131] yes The transposed matrix, ω i,1,l represents the radial basis function, τ i,1 represents the approximation error, and τ i,1 >0. R l and R n Denote the l-dimensional and n-dimensional real number sets respectively. Introduce an unknown positive parameter Q i,1 ,definition Unknown parameter Q i,1 Can be achieved through Estimation, the estimation error is defined as

[0132] S022. Design virtual control law i,1 as follows:

[0133]

[0134] where μ i,1 is a positive design parameter, β zi,1 is a greater than |z i,1 | a positive number, It's Q i,1 estimated value.

[0135] Designing adaptive control laws for:

[0136]

[0137] in φ i,1 , q i,1 and δ i,1 These are all positive design parameters.

[0138] S03. Design a relative threshold event trigger mechanism to reduce the signal update frequency;

[0139] In order to improve the utilization rate of communication resources and save communication resources, a relative threshold event triggering mechanism is designed. The mathematical model expression of the event triggering mechanism is as follows:

[0140]

[0141] Among them, ω i,j (t), j=1,…,m represents event-triggered control input, λ>0,0<ε<1, are all positive design parameters. dj Indicates the control signal, Indicates the difference between the control signal and the actuator input. When satisfied, the control input signal Will be updated, then there is t k+1 is the triggering moment, t k Indicates the last trigger moment. Otherwise

[0142] Assuming that the maximum number of failed actuators is m-1, the remaining actuators can still achieve the control target. To further illustrate the problem of actuator failure, the following definition is made:

[0143] The moment when the j1th actuator fails is defined as but The time interval from the j1th actuator failure to the j1+1th actuator failure. is the set of actuators that fail completely, is the remaining set of executors, then

[0144] Designed control signal u dj The construction is:

[0145]

[0146] in and η i =[υ i,2 η i,1… η i,m ] are in time intervals middle, Represents ρ i,j If the transposed matrix of That is, when the j2th actuator fails, there is otherwise c i,j is an unknown parameter, υ i,2 This is the second virtual control law, which will be designed next.

[0147] Therefore, the following results can be obtained:

[0148]

[0149] Derived:

[0150]

[0151] The above formula shows that, by design, regardless of whether the actuator fails, the input of the manipulator can be expressed as the second virtual control law υ i,2 At this time, by designing a reasonable virtual control law i,2 This can ensure the control performance of the system when the actuator fails. However, it should be noted that u dj c in i,j ,θ i,j and h i,j | is unknown, so ρ i,j It cannot be obtained in advance and needs to be As ρ i, The estimated value of j, its estimated estimation error Therefore, u dj The expression is:

[0152]

[0153] S04, through the adaptive law and The second virtual control law υ is derived i,2 ,υ i,2 Used to approximate unknown system parameters online to control multi-link robotic arms;

[0154] Approximate the uncertain part of the system through neural network; according to the virtual error z i,2 Design virtual control law i,2 , and design the adaptive law and Furthermore, the step S04 specifically includes:

[0155] S041. Approximating the uncertain combination part in the system model through radial basis function neural network:

[0156]

[0157] Therefore, the approximation process The expression is as follows:

[0158]

[0159] in, represents the ideal weight vector, Yes i,2 The transposed matrix of yes The transposed matrix, ω i,2,l represents the radial basis function. τ i,2 represents the approximation error, and τ i,2 > 0. Introducing an unknown positive parameter Unknown parameter Q i,2 Can be achieved through The estimated error is defined as:

[0160] S042, according to the second error variable z i,2 and design the second virtual control law υ using the backstepping design method and the barrier Lyapunov function i,2 .

[0161] Design virtual control law i,2 as follows

[0162]

[0163] in q i,2 , and μ i,2 are all positive design parameters, is a greater than |z i,2 |A positive number. It's Q i,2 estimated value.

[0164] Designed adaptive law and for:

[0165]

[0166]

[0167] Where 0<σ<1, φ i,2 , χ i,j , δi,2 and are all positive design parameters, is ρ i,j estimated value.

[0168] Stability analysis:

[0169] First, select the barrier Lyapunov function V i,1 and V i,2 It can be expressed as: According to theoretical analysis, for any 0<Λ<1, there exists a positive constant Where C and D are constants, let when When combining the designed virtual control law and adaptive control law, according to theoretical analysis, we can get:

[0170] V σ <ε

[0171] From the above formula, we can infer that V is bounded, then z i,1 ,z i,2 , and are all bounded. In addition, Q i,1 ,Q i,2 and ρ i,j are all constants, so and are all bounded. At the same time, because y d and z i,1 is bounded, indicating that y i is also bounded. Therefore, the components of i,1 and υ i,2 The parameters of are bounded, indicating that υ i,1 and υ i,2 It is also bounded, and we can deduce that x i,1 and x i,2 are all bounded, which means that the control law u dj In summary, we can get that all the outputs and tracking errors of the closed-loop system are bounded, and we can also get that when The control method proposed in this invention can make the output y of each robot arm i can consistently track the trajectory of the virtual leader y d .

[0172] In one embodiment, the control method is simulated and verified based on the Matlab experimental platform. Figure 2The communication topology with one virtual leader and four followers is described. Nodes 1-4 represent follower manipulators, and node d represents the virtual leader. For the sake of generality, the output signal (reference signal) of the virtual leader is assumed to be y d =sin(2t).

[0173] In order to deal with the uncertain part of the system, a radial basis function neural network is used, and its Gaussian function is expressed as follows:

[0174]

[0175]

[0176] in l=1,2,...,9,κ 1,1,l =κ 2,2,l =κ 3,2,l =[-5+l,-5+l,-5+l,-5+l] T , κ 2,1,l =κ 3,1,l =[-5+l,-5+l,-5+l] T , κ 1,2,l =κ 4,1,l =[-5+l,-5+l,-5+l,-5+l,-5+l] T , κ 4,2,l =[-5+l,-5+l,-5+l,-5+l,-5+l,-5+l] T .

[0177] The parameters of the robotic arm system are: mass M = 1 kg, viscous friction coefficient B = 1 Pas, gravity acceleration G = 9.8 m / s 2 , the length of the middle axis L = 0.4m, the moment of inertia The system input disturbance F1 = sin(t). Due to practical limitations, the state of the system needs to satisfy |x i,1 |<1.52、|x i,2 |<3.5. Figure 2 It can be seen that b1=1, b2, b3 and b4 are all 0, and the adjacency matrix is ​​as follows:

[0178]

[0179] The controller parameters are selected as follows: μ 1,1 =55,μ 2,1 =40, μ 3,1 =25,μ 4,1 =45,μ 1,2 =15,μ 2,2 =35,μ3,2 =15,μ 4,2 =25, σ=0.9, φ1=0.1, φ2=0.1, δ1=0.1, δ2=0.01, q1=1.35,q2=1.36, χ1=χ2=0.01,

[0180] The parameters of the event trigger controller are: λ = 2, ε = 0.1, ε1 = 0.1,

[0181] Consider the following two actuator failure scenarios:

[0182] Case 1: The first actuator in each robotic arm is in normal working condition, while the second actuator in each robotic arm has a 40% failure at t = 16.5s.

[0183] Case 2: When t = 16.5s, the first actuator in each robotic arm fails by 20%, and the second actuator fails completely. The outputs of the second actuators of the four robotic arms become 0.1, 0.2, 0.1, and 0.3, respectively.

[0184] The simulation results for the first case are as follows Figures 3 to 8 As shown in the figure, the simulation results of case 2 are as follows Figures 9 to 14 shown.

[0185] Depend on Figure 3-38 It can be seen that all signals are bounded. Figure 3 and Figure 21 The output signals of the leader and follower under two actuator failure conditions are shown respectively, indicating that the output signals of all manipulators have good consistent tracking performance; the synchronization error curve is shown in Figure 4 and Figure 22 As shown in Figure 2, it can be seen that the synchronization error enters the 5% error band in a very short time, which shows that the system converges very quickly. The status of each robot arm is as follows: Figure 5-8 and Figure 23-26 As shown in the figure, it can be seen that the states of each robot arm can achieve good consistency; the input signal is as follows Figure 9-20 、 Figures 27-38 As shown in , the event-triggered control input ω(t) is continuous and smooth, while the control input u(t) is sawtooth. The event triggering time interval of each follower manipulator is as follows: Figure 8 and Figure 14As shown in Table 1, the minimum trigger interval for all follower manipulators is greater than 0.01s, preventing the phenomenon of infinite triggering within a limited time. Table 1 shows that each manipulator saves a significant amount of communication resources. Table 2 shows that compared to the traditional time-triggered mechanism, the event-triggered mechanism significantly reduces the number of triggers, lowering the signal update frequency and saving communication resources.

[0186] Table 1 Resource saving rate

[0187]

[0188] Table 2 Triggering times of each agent

[0189]

[0190] From the simulation results, it can be seen that when actuator failures and state constraints occur in a multi-link robotic arm system, this method achieves state synchronization. Each follower robotic arm can well track the trajectory of the leader robotic arm, and the synchronization error is kept within a very small range, while saving a lot of communication resources.

[0191] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit the scope of protection of the present invention. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that the technical solutions of the present invention may be modified or replaced by equivalents without departing from the essence and scope of the technical solutions of the present invention.

Claims

1. A finite-time control method for consistency tracking of a multi-link manipulator, characterized in that: The following steps are involved: S01: Establish a multi-link robot arm model and an actuator fault model, and combine the two to obtain a dynamic model of the multi-link robot arm system with actuator faults; S02: Design the consistent tracking error system of the multi-link manipulator system and derive the first virtual control law υ i,1 and adaptive law S03. Design a relative threshold event trigger mechanism to reduce the signal update frequency; S04, through the adaptive law and The second virtual control law υ is derived i,2 ,υ i,2 Used to approximate unknown system parameters online to control multi-link robotic arms; In S01, the dynamic model of the multi-link robotic arm system is described by the following differential equation: Among them, i∈{1, 2, 3, 4}, represents i robotic arms; x i，1 and x i，2 are the displacement and velocity of the single-link manipulator, and x i，1 and x i，2 The derivative of u i and y i Represent the input and output of the i-th manipulator, J is the inertia of the single-link manipulator, M is the mass of the single-link manipulator, B is the viscous friction coefficient, G is the acceleration of gravity, L is the length between the joint and the center of mass, and function F1 represents the input disturbance; and x i，1 <1.52, x i，2 <3.5; In S02, the consistency tracking error system of the multi-link manipulator system is defined by the following formula: Where z i,1 ,z i,2 is the error variable, υ i,1 is the first virtual control law, l i is the transfer function, E i is the synchronization error of the i-th robot arm, Among them, y d is the trajectory of the virtual leader, y k represents the output of the k-th robotic arm, b i is the weight of the edge from the i-th robot to the leader. When the robot i receives the information from the leader, b i > 0, otherwise, b i =0;a ik Represents the weight of the edge from the i-th robotic arm to the k-th robotic arm. When the information of the k-th robotic arm can be transmitted to the i-th robotic arm, a ik > 0, otherwise a ik =0; The first virtual control law υ i,1 and adaptive control law as follows: where μ i,1 and σ are both positive design parameters, where 0<σ<1, is a greater than |z i,1 | a positive number, For unknown weight vector It can be estimated using the adaptive law, assuming It's Q i,1 The estimated value of in φ i,1 , q i,1 and δ i,1 are all positive design parameters, z i,1 represents the error variable, ω i,1,l represents radial basis function.

2. The consistency tracking finite time control method of a multi-link robot arm according to claim 1 is characterized in that: In the above S03, in order to improve the utilization rate of communication resources and realize the conservation of communication resources, a relative threshold event triggering mechanism is designed. The mathematical model expression of the event triggering mechanism is as follows: Among them, ω i,j (t) represents the event-triggered control input, h i,j represents the unknown parameters in the system, γ i,j (x i ) represents the known nonlinear function in the system, ε, ε1, and λ are both positive design parameters, λ>0,0<ε<1, u dj Indicates the control signal, Indicates the difference between the control signal and the actuator input; only when the pre-designed trigger condition When satisfied, the control input signal Will be updated, then there is t k+1 is the triggering moment, t k Indicates the last trigger moment, otherwise R represents the set of real numbers.

3. The consistency tracking finite time control method of a multi-link robot arm according to claim 2 is characterized in that: In the S04, the second virtual control law υ i,2 , adaptive control law and Satisfies the following calculation formula: in, q i,2 , and μ i,2 are all positive design parameters, ω i,2,l represents the radial basis function, z i,2 represents the error variable, is a greater than |z i,2 | a positive number, It's Q i,2 estimated value of; Where 0<σ<1, φ i,2 , χ i,j , δ i,2 and are all positive design parameters, is ρ i,j estimated value.

Images