A directed graph-based distributed fixed-time consensus tracking control method for multi-robot systems

Abstract

The application provides a kind of directed graph-based multi-robot arm system distributed fixed time consistent tracking control method. With the development of robot arm in the industrial field, the cooperation of multi-robot arm is the future trend, and the cooperative control algorithm is very important for multi-robot arm system to complete the cooperative task. Based on the backstepping technique, the communication between subsystems is described by directed graph, and the unknown nonlinear characteristics are approximated by radial basis function neural networks (RBFNNs); secondly, in view of the network congestion problem that may exist in actual use, an event-triggered mechanism is designed to reduce the update frequency of the control signal to alleviate the communication pressure; thirdly, based on the fixed time stability theory, a distributed event-triggered consensus tracking controller is constructed. In theory, the synchronization error can converge to the vicinity of the origin within a time independent of the initial state of the system. Simulation experiments show that the system is stable and all follower robot arms consistently track the output trajectory of the leader.

Description

TECHNICAL FIELD

[0001] The present application belongs to the field of mechanical arm control, and particularly relates to a directed graph-based distributed fixed-time consensus tracking control method for a multi-robot arm system. BACKGROUND

[0002] With the development of mechanical arms in the industrial field, the work tasks become more and more complex, and a single mechanical arm cannot be well completed, and the cooperation of multiple mechanical arms is the future trend. In a multi-robot arm system, all subsystems are connected through a communication topology network, and each subsystem completes the cooperative task by sensing the state information of the neighbors or leaders. The multi-robot arm system needs the support of cooperative control algorithm to complete the cooperative task, so it is of great significance to study the cooperative control algorithm of multi-robot arm.

[0003] In the current research, the cooperative control of multi-robot arm system is not perfect. Some scholars use graph theory to describe the information transmission inside the multi-robot arm system and propose a consensus tracking control design method. Some scholars introduce the command filter method into the backstepping process and propose an adaptive command filter backstepping control method for multi-robot arm system. Although these methods can realize the cooperative control of multi-robot arm, these methods are non-fixed time control methods, and the realization of the final stable state of the system theoretically needs an infinite long time. On the other hand, the realization of the cooperative control of the multi-robot arm system needs high-frequency continuous communication between the internal systems, but the communication resources of the system are limited, and these methods do not consider the communication resource constraints that may be faced in actual application. SUMMARY

[0004] The main purpose of the present application is to provide a directed graph-based distributed fixed-time consensus tracking control method for a multi-robot arm system, which aims to solve the defects existing in the background technology.

[0005] To achieve the above purpose, the present application provides a directed graph-based distributed fixed-time consensus tracking control method for a multi-robot arm system, which comprises the following steps:

[0006] S1: considering a multi-robot arm system with 1 leader and N followers, all the robot arms are connected through a directed communication topology network, and a robot arm can only obtain all the state information of its neighbors or leaders. The leader is denoted as "0" and the follower is denoted as "k" (k=1, 2,..., N). The system dynamics model of the follower is established.

[0007] S2: define the synchronization error z of the follower k , the virtual control error e k , and construct the virtual control law a k,i based on the first preset Lyapunov function.

[0008] S3: Design event-triggered mechanism and generate virtual control law based on the second preset Lyapunov function by using RBFNNs

[0009] S4: Design adaptive law based on the third preset Lyapunov function

[0010] S5: Simulation analysis. All signals of the closed-loop system are bounded, and the joint angles of all followers can well track the output angles of the leader, and no Zeno phenomenon occurs.

[0011] The application provides a directed graph-based distributed fixed-time consensus tracking control method for a multi-robot arm system, has the following beneficial effects: the desired trajectory can be well tracked, the system can realize rapid convergence in different initial states, no Zeno phenomenon occurs, and the collaborative control of the nonlinear multi-robot arm system can be well realized, and in addition, the method effectively saves system communication resources. BRIEF DESCRIPTION OF DRAWINGS

[0012] Figure 1 is a controller design block diagram;

[0013] Figure 2 is a communication topology diagram of the multi-robot arm system;

[0014] Figure 3 is the output of each follower joint 1;

[0015] Figure 4 is the output of each follower joint 2;

[0016] Figure 5 is the synchronization error of each follower joint 1;

[0017] Figure 6 is the synchronization error of each follower joint 2;

[0018] Figure 7 is the event time interval of the follower 1;

[0019] Figure 8 is the event time interval of the follower 2;

[0020] Figure 9 is the event time interval of the follower 3; DETAILED DESCRIPTION

[0021] The embodiments of the application will be described in detail below. The following embodiments are implemented on the premise of the technical scheme of the application, and detailed implementation modes and specific operation processes are given, but the protection scope of the application is not limited to the following embodiments.

[0022] A directed graph-based multi-robot system distributed fixed-time consensus tracking control method, comprising the following steps:

[0023] S1: considering a multi-robot system with 1 leader and N followers, all the robots are connected through a directed communication topology network, and one robot can only obtain all the state information of its neighbors or the leader. The leader is denoted as "0" and the follower is denoted as "k" (k = 1, 2,..., N). The system dynamics model of the follower is established:

[0024]

[0025] where q k = [q k,1 , q k,2 ,..., q k,n ] T is the system output joint angle, is the angular velocity, is the angular acceleration; M k (q k ) is the inertia matrix, is the centrifugal force and Coriolis matrix, G k (q k ) is the gravity vector, represents the friction matrix, u k = [u k,1 , u k,2 ,..., u k,n ] T is the input torque.

[0026] S2: define the synchronization error z k of the follower, the virtual control error e k , and construct the virtual control law a k,i based on the first preset Lyapunov function.

[0027] Specifically, the synchronization error of the follower k is defined as follows:

[0028]

[0029] where z k = [z k,1 , z k,2 ,..., z k,n ] T is the position synchronization error of the follower k; q0 = [q 0,1 , q 0,2 ,..., q 0,n ] T is the output of the leader; t k> 0 is the information transmission coefficient between the follower k and the leader; M k is the neighbor set of the follower k; q j = [q j,1 , q j,2 ,..., q j,n ] T is the output of the neighbor j; β kj is the information transmission coefficient between the follower k and the neighbor j.

[0030] Further, the virtual control error is defined as follows:

[0031]

[0032] wherein e k = [e k,1 , e k,2 ,..., e k,n ] T is the virtual control error, a k = [a k,1 , a k,2 ,..., a k,n ] T is the virtual control law to be designed.

[0033] The first preset Lyapunov function in S2 is as follows:

[0034]

[0035] The virtual control law a k,i is designed as follows:

[0036]

[0037]

[0038] wherein b k,1i and c k,1i are positive design parameters,

[0039] S3: based on the second preset Lyapunov function, an event-triggered mechanism is designed by using RBFNNs, and a virtual control law is generated

[0040] The second preset Lyapunov function in S3 is as follows:

[0041]

[0042] After derivation on the selected Lyapunov function, there is an uncertain function F k (X k ), wherein,

[0043]

[0044] F k,i is approximated by RBFNNs k,i ) as:

[0045]

[0046] where Φ k,i (X k,i ) = [Φ k,i,1 (X k,i ), Φ k,i,2 (X k,i ),..., Φ k,i,N (X k,i )] T ∈ R N is the known basis function vector, N is the number of neurons, is the unknown ideal weight vector, ε k,i (X k,i ) ∈ R is the bounded approximation error, i.e.

[0047] According to Young's inequality, we have:

[0048]

[0049]

[0050] where a k,i > 0, is an unknown constant, and ||·|| denotes the standard two-norm.

[0051] Then the triggering mechanism is designed as follows:

[0052]

[0053]

[0054] where m k,i (t) = ω k,i (t) - u k,i (t) is the measurement error, is the virtual control law to be designed, and the related design parameters satisfy δ k,i > 0, 0 < p k,i < 1, s k,i > 0; t k,i,q represents the time when an event is triggered, q ∈ Z + , and t k,i,1 is the initial time.

[0055] Further, a virtual control law is designed As follows:

[0056]

[0057] where b k,2i > 0, c k,2i > 0, is the estimated value of θ k,i .

[0058] S4: Based on the third preset Lyapunov function, an adaptive law is designed

[0059] The third preset Lyapunov function in S4 is as follows:

[0060]

[0061] where r k,i > 0.

[0062] Derive V k,3 and design an adaptive law As follows:

[0063]

[0064] where λ k,1i > 0, λ k,2i > 0

[0065] S5: Simulation analysis.

[0066] Further, we consider a multi-robot system with 1 leader and 3 followers, Figure 2 The communication topology of the system is described, where "0" represents the leader and "1-3" represents the followers. It is assumed that the output of the leader is q 0,1 = sin(2t), q 0,2 = sin(2t). According to Figure 2 , the adjacency matrix τ = diag{1, 0, 0.8} and

[0067] The dynamics model of robot k (k = 0, 1, 2, 3) is as follows:

[0068]

[0069] where,

[0070] C k,22 = 0,

[0071] G k (q k ) = [G k,1 G k,2 ] T G k,1 =(m k,1 l k,c2 +m k,2 l k,1 )gcosq k,1 +m k,2 l k,c2 gcos(q k,1 +q k,2 ), G k,2 =m k,2 l k,c2 gcos(q k,1 +q k,2 ),

[0072] u k =[u k,1 ,u k,2 ] T .

[0073] m k,1 and m k,2 The masses of connecting rod 1 and connecting rod 2 are respectively, l k,1 and l k,2 The length of the link.

[0074] I k,1 and I k,2 Let l be the moment of inertia. k,c1 and l k,c2 Let q1 be the position of the center of mass of the two connecting rods. The initial state of the system is q1 = q2 = q3 = [0.10, 0.10]. T rad,

[0075] The controller is designed as follows:

[0076]

[0077]

[0078] Where i = 1, 2, j = 1, 2, ..., 16, k = 1, 2, 3, and the simulation step size is 0.01s.

[0079] From the center C of the Gaussian function k,i,j The matrix C formed k,i =[C k,i,1 C k,i,2 ,...,C k,i,16 The design is as follows:

[0080] For follower 1, since it can only get the state information of leader 0, so The rest of the parameters are chosen as follows: b 1,11 = c 1,11 = 10, b 1,21 = c 1,21 = 0.1, b 1,12 = c 1,12 = 5, b 1,22 = c 1,22 = 0.1, a 1,1 = a 1,2 = 0.5, r 1,1 = r 1,2 = λ 1,11 = λ 1,12 = λ 1,21 = λ 1,22 = 0.1, p 1,1 = p 1,2 = 0.1, s 1,1 = s 1,2 = 2, d 1,1 = d 1,2 = 1.

[0081] Follower 2 can only get the state information of follower 1, so The rest of the parameters are chosen as follows:

[0082] b 2,11 = c 2,11 = 11, b 2,21 = c 2,21 = 0.01, b 2,12 = c 2,12 = 5, b 2,22 = c 2,22 = 0.01, a 2,1 = a 2,2 = 0.5, r 2,1 = r 2,2 = λ 2,11 = λ 2,12 = λ 2,21 = λ 2,22 = 0.1, p 2,1 = p 2,2 = 0.1, s 2,1 = s 2,2 = 2, d 2,1 = d 2,2 = 1.

[0083] Follower 3 can only get the state information of leader 0, so The remaining parameters are selected as follows: b 3,11 =c 3,11 =8, b 3,21 =c 3,21 =0.1, b 3,12 =c 3,12 =5, b 3,22 =c 3,22 =0.01, a 3,1 =a 3,2 =0.5,,r 3,1 =r 3,2 =λ 3,11 =λ 3,12 =λ 3,21 =λ 3,22 =0.1,ρ 3,1 =ρ 3,2 =0.1,s 3,1 =s 3,2 =2,δ 3,1 =δ 3,2 =1.

[0084] from Figures 3 to 9 It can be seen that all signals in a closed-loop system are bounded. From Figure 3 and Figure 4 It can be seen that the two joint angles of all follower robotic arms can track the desired output angle very well. Figure 5 and Figure 6 The synchronization error of the joint positions in the multi-arm robotic system is as follows: the synchronization error of follower robotic arms 1 and 3 can be controlled within ±0.05 rad; follower robotic arm 2 receives the status signal of robotic arm 1. Since the signal itself fluctuates greatly, the synchronization error is difficult to control, so the cumulative error is large. Figures 7 to 9 The data describes the trigger event time intervals for all follower robotic arms, with a maximum interval of 0.34s and a minimum interval of 0.01s, thus eliminating the Zeno phenomenon. Finally, compared to traditional time-triggered control methods, the number of event triggers for both methods was statistically analyzed. The method provided by this invention saves 64.4% of communication resources.

[0085] The preferred embodiments of the present invention have been described in detail above. It should be understood that those skilled in the art can make numerous modifications and variations based on the concept of the present invention without creative effort. Therefore, all technical solutions that can be obtained by those skilled in the art based on the concept of the present invention through logical analysis, reasoning, or limited experimentation on the basis of existing technology should be within the scope of protection defined by the claims.

Claims

1. A directed graph-based multi-manipulator system distributed fixed-time consensus tracking control method, characterized in that, Comprising the following steps: S1: considering a multi-robot system with 1 leader and N followers, all robots are connected through a directed communication topology network, one robot can only obtain all state information of its neighbors or the leader, where the leader is denoted as "0" and the follower is denoted as "k" (k = 1, 2,..., N), and a system dynamics model of the follower is established; S2: define the synchronization error z of the follower k and the virtual control error e k , construct the virtual control law a based on the first preset Lyapunov function k,i ; In the step S2, the synchronization error of the follower k is as follows: where q k = [q k,1 , q k,2 ,..., q k,n ] T is the system output joint angles, z k = [z k,1 , z k,2 ,..., z k,n ] T is the position synchronization error of the follower k, q0= [q 0,1 , q 0,2 ,..., q 0,n ] T is the output of the leader, τ k > 0 is the information transmission coefficient between the follower k and the leader, M k is the neighbor set of the follower k, q j = [q j,1 , q j,2 ,..., q j,n ] T is the output of the neighbor j, β kj is the information transmission coefficient between the follower k and the neighbor j; In the step S2, the virtual control error is defined as follows: wherein, is the angular velocity, e k = [e k,1 , e k,2 ,..., e k,n ] T is the virtual control error, a k = [a k,1 , a k,2 ,..., a k,n ] T is the virtual control law; In the step S2, the first preset Lyapunov function is designed as follows: In the step S2, the virtual control law a k,i The design is as follows: where b k,1i and c k,1i are positive design parameters, M k is the neighbor set of follower k; β kj is the information transfer coefficient between follower k and neighbor j; S3: Based on the second preset Lyapunov function, an event-triggered mechanism is designed by using RBFNNs, and a virtual control law is constructed In the step S3, the second preset Lyapunov function is as follows: In the step S3, the event-triggered mechanism is designed as follows: wherein m k,i (t) = ω k,i (t) - u k,i (t) is the measurement error, is the virtual control law to be designed, and the relevant design parameters satisfy δ k,i > 0, 0 < p k,i < 1, s k,i > 0, t k,i,q represents the time of triggering of a certain event, q e Z + , t k,i,1 is the initial time; In the step S3, the virtual control law The design is as follows: where a k,i > 0, b k,2i > 0, c k,2i > 0, is an estimate of θ k,i , Φ k,i (X k,i ) = [Φ k,i,1 (X k,i ), Φ k,i,2 (X k,i ),..., Φ k,i,N (X k,i )] T ∈ R N denotes a known basis function vector, N is the number of neurons, and ||·|| denotes the standard two-norm. S4: design an adaptive law based on the third preset Lyapunov function In the step S4, the third preset Lyapunov function is designed as follows: wherein r k,i >0; In the step S4, the adaptive law The design is as follows: wherein a k,i > 0, λ k,1i > 0, λ k,2i > 0; S5: simulation analysis is performed on the proposed control method.

2. The directed graph-based distributed fixed-time consensus tracking control method for multi-robot systems of claim 1, wherein, In the step S1, the established dynamics model of the follower is as follows: where q k = [q k,1 , q k,2 ,..., q k,n ] T is the system output joint angles, is the angular velocity, is the angular acceleration; M k (q k ) is the inertia matrix, is the centrifugal and Coriolis force matrix, G k (q k ) is the gravity vector, represents the friction matrix, u k = [u k,1 , u k,2 ,..., u k,n ] T is the input torque.

3. The directed graph-based distributed fixed-time consensus tracking control method for multi-robot systems of claim 1, wherein, In the step S3, after derivation of the selected Lyapunov function, there is an indeterminate function F k (X k ), wherein, τ k >0 is the information transmission coefficient between the follower k and the leader, M k is the neighbor set of the follower k; β kj is the information transmission coefficient between the follower k and the neighbor j; Using RBFNNs for F k,i (X k,i ) approximation process gives: where Φ k,i (X k,i ) = [Φ k,i,1 (X k,i ), Φ k,i,2 (X k,i ),..., Φ k,i,N (X k,i )] T ∈ R N represents a known basis function vector, N is the number of neurons, represents an unknown ideal weight vector, ε k,i (X k,i ) ∈ R is a bounded approximation error, i.e.:

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