Distributed Iterative Learning Formation Control for Nonholonomic Multiple Wheeled Mobile Robots with Channel Noise

In this paper


Introduction
In recent years, WMR has been widely used in the logistics industry, military reconnaissance, agricultural transportation and other practical projects [24], so the control of WMR has been rapidly developed, such as sliding mode control [27], backstepping control [7], adaptive control [15], feedback control [19] and observer-based control [1], etc.
Due to the limited performance of a single WMR, it is difficult to complete a large project alone, so multiple WMRs cooperative control is more practical.More and more scholars have done a lot of research on the cooperative control of multiple WMRs, where formation control is one of the important research directions.Formation control is a coordinated movement that a group of similar WMRs formed and maintain a certain formation in the process of movement.Traditional formation research methods mainly include leader-following formation [3], behavior-based method [2], virtual structure method [11] and based on graph theory distributed formation method [23].Shao et al. [21] proposed a three-layer hybrid control architecture for controlling groups of mobile robots, to achieve formation based on the leader-following scheme.Ou et al. [16] transformed the tracking error of the robot into two subsystems and designed a continuous distributed controller for each robot by analyzing the two subsystems respectively, the group of WMRs can track the reference trajectory was achieved.In WMR formation control, battery power and the processor's computing ability is limited, so how to design more efficient communication and control methods between the WMRs to become a challenging problem to be solved.
The distributed control scheme allows computing work to be divided among all WMRs in the formation, each WMR sharing information with its leader only via wireless network, so the design of distributed formation control scheme becomes an important research direction.In [10] and [18], the consensus information is used to construct the formation error and design the distributed formation algorithm.However, all of the above control approaches are asymptotic stable control along the time domain.Although stable formation can be achieved, the transient performance of the system is poor at the initial stage.
In addition, there is a class of tasks that are run repeatedly with finite time interval in some practical applications of WMR.Most of the existing formation control algorithms are asymptotically stable control in time domain, but the transient performance was not ideal.Iterative learning control (ILC) is an effective solution for single WMR tracking trajectory in repetitive environment [22], which can achieve the complete tracking in time domain.ILC has been widely used in the field of collaborative control [6,13,14].Liu and Jia [14] studied the formation problem for discrete-time multi-agent systems with unknown nonlinear dynamics, a distributed iterative learning method was proposed based on the local formation error data and the sufficient conditions for preserving formation was deduced.The accurate formation was guaranteed in the presence of initial formation errors and switching topology network.Li and Li [13] considered the consensus tracking problem for a class of nonlinear multi-agent systems, an adaptive ILC consensus algorithm was developed, where the convergence condition was given based on the Lyapunov stability theory.Bu et al. [6] provided a model-free adaptive iterative learning consensus tracking strategy by using the dynamical linearization technique along with the iteration domain, to settle the consensus tracking problem for nonlinear multiagent systems.However, the above collaborative and formation algorithms were designed on the foundation of multi-agent systems.Since WMR is a typical nonlinear system with non-holonomic constraints, these algorithms cannot be directly applied to WMRs formation, so it is necessary to study multiple WMRs distributed formation algorithms based on ILC.
In practice, the data transmission of wireless communication network between WMRs is not always correct, which makes it more difficult to design and analyze the convergence of the controller.To solve the problem of ILC with network constraints, a series of solutions have been proposed in [4,5,8,9,12,20,26].Zhang and Li [26] introduced the Sigma Delta quantizer with limited communication data rate into the control system, a quantized iterative learning control was designed and the analysis of convergence utilized a decreasing learning gain.to solve the tracking problem of continuous-time multi-agent system in the case of quantization and packet losses.Bu et al. [5] studied the robustness of the ILC algorithm for nonlinear systems with random measurement disturbance and proposed an improved iterative learning methodology of gain reduction based on p-type ILC, which can guarantee that the tracking error of the nonlinear system in presence of measurement disturbances to zero.However, unlike measurement noise and state disturbance, channel noise is introduced externally.Because of the different positions it produces, channel noise can be divided into sensor-controller (SC) noise and controller-actuator (CA) noise.The former arises when exchanging measurement data between the sensor and the controller, and involves itself in the learning process.The latter arises when exchanging input data between the controller and the actuator, and affects the convergence of output error directly.Huang and Fang [8] studied the channel noise in ILC, and it was found that the contributions of SC and CA noise to errors were affected by the learning gain.Huang et al. [9] proposed an adaptive learning gain ILC to improve the convergence performance of the linear system with channel noise.However, WMR as a typical nonlinear system, the above strategy cannot be directly expanded to multiple WMRs formation control with channel noise.
Based on the above analysis, this paper designs a distributed iterative learning formation control algorithm for formation problem of multiple WMRs, and explores the convergence problem of the system under the channel noise and switching topologies.Compared with [26] and [5], this paper mainly analyzes the formation system of multiple mobile robots, which is more complex than multi-agent system.Compared with [9], this paper solves the convergence problem of channel noise in the nonlinear system of mobile robots and extends it to formation motion.The main contributions of this paper are as follows: 1 Compared with the traditional multiple WMRs formation algorithm, the proposed algorithm is based on the iteration axis instead of the time axis.The transient performance is better because it can track each time point in a limited running interval.
2 Based on the nonlinear system model of mobile robot, the consensus error is constructed.The current and predicted errors are introduced into the controller as learning items to improve the formation effect of the system under channel noise.The upper bound of formation error is obtained and the effect of channel noise on the formation tracking effect of multiple WMRs is analyzed.
The rest of this paper is organized as follows.In section 2, the necessary preliminary work and the control objective are given respectively.In section 3, the formation tracking controller is designed, and the influence of channel noise on formation tracking is analyzed.The numerical simulation is given in section 4.
Finally, we conclude this paper in section 5.

Preliminaries
In this paper, R represents the set of real numbers,

|| ||
A represents a matrix norm for the matrix n n A R × ∈ , {} diag ⋅ represents the diagonal matrix and I represents the identity matrix.On algebraic graph theory, the communication topology between WMRs can be described by a mathematical matrix.The weighted directed graph is denoted as is the set of vertexes, it represents the set of robots in this paper.E V V ⊆ × is the set of edges.{ | ( , ) } i N j V j i E = ∈ ∈ denotes the neighborhood of robot i. … is called the in-degree matrix and A directed graph is said to be strongly connected if a path exists between any two vertexes in the graph.Assume that the directed graph contains at least one spanning tree, with node 0 as the root.

Problem Formulation
We consider a group of N WMRs, each robot has the same mechanical structure as shown in Fig. 1.
, , , ( ), ( ), ( ) x t y t t θ are the generalized position and .It is assumed that the wheels of the robot and the ground cannot move in a direct sideslip, then WMR has the following non-holonomic constraint, , .
Under the non-holonomic constraint, the discrete-time model of the WMR i at the k-th iteration in the form as follows: v t w t are linear and angular velocities respectively.
The states matrix, velocities matrix and control input matrix of WMR i are represented by , , , ( ), ( ), ( ( )) Then equation ( 2) can be rewritten as follows: The non-holonomic WMR model (3) adopted in this paper satisfies the following properties: Property 1: The nonlinear matrix function , and b l is a positive constant.
Property 2: the matrix function In this paper, the reference trajectory d η is regarded as the running track of a virtual leader WMR, and the virtual leader is denoted as number 0. Defining , where || ( ) || The goal of formation control in this paper is designing a suitable control algorithm for the velocity ( ) k u t of WMR, so that all WMRs can maintain the desired formation during the tracking along the reference track.For all According to the rules of communication transmission between adjacent WMRs, the consensus error is given: where , i j a A ∈ express the communication of the WMR i and j in the adjacency matrix.

Configuration of WMR
The states matrix, velocities matrix and control input matrix of WMR i are represented by , , , ( ), ( ), ( ( )) Then equation ( 2) can be rewritten as follows: The non-holonomic WMR model (3) adopted in this paper satisfies the following properties: reference trajectory, otherwise 0 i s = .We assume that the reference trajectory ( 1) d t η + is known, there is a bounded expected control input ( ) where || ( ) || The goal of formation control in this paper is designing a suitable control algorithm for the velocity ( ) k u t of WMR, so that all WMRs can maintain the desired formation during the tracking along the reference track.
According to the rules of communication transmission between adjacent WMRs, the consensus error is given: where , i j a A ∈ express the communication of the WMR i and j in the adjacency matrix.When WMRs transmit information over the wireless network, the input , ( ) i k u t and output error , ( ) i k t δη need to be transmitted from the controller to the actuator and form the sensor to the controller respectively.The network model is shown in Fig. 2.

Figure 2
The WMR system transmits information over wireless networks by ( ), ( ) m t n t respectively, then the tracking error and the input data can be represented as then the consensus error can be rewritten as Remark 1: Due to the unreliability of the wireless network, the output errors , ( ) i k t δη received by the controller and input data , ( ) i k u t obtained by the actuator will be mixed with the channel noise during the transmission.

Figure 2
The WMR system transmits information over wireless networks

Control Algorithm Design and Convergence Analysis
With the above consensus error, the DILFC algorithm as shown in Fig. 3 is proposed: To facilitate the analysis, we have the following assumptions: Assumption 1: The initial state of each WMR is the same in every iteration and , (0) (0) Assumption 2: The channel noise ( ), ( ) m t n t are bounded for every iteration, that is Assumption 3: Each switching communication graph has a spanning tree with a virtual leader as the root.
Remark 2： Assumption 1 is a basic condition of the ILC system, the identical initial condition is still followed in this paper.Furthermore, in [9], channel noise is assumed to be bounded, and the same assumption is still adopted in this paper.So, Assumption 1 and Assumption 2 are reasonable.

Figure 3
Block diagram of the scheme performed for WMR For convenience, we define , 1, 2, , i i N δ = … as the i th eigenvalue of L S + .Based on these assumptions, the following theorem gives the main results of this paper.

Theorem: Considering multiple WMRs satisfies assumptions 1~3, the DILFC controller is
The CA and SC channel noise are represented by ( ), ( ) m t n t respectively, then the tracking error and the input data can be represented as by ( ), ( ) m t n t respectively, then the tracking error and the input data can be represented as then the consensus error can be rewritten as Remark 1: Due to the unreliability of the wireless network, the output errors , ( ) i k t δη received by the controller and input data , ( ) i k u t obtained by the actuator will be mixed with the channel noise during the transmission.

Figure 2
The WMR system transmits information over wireless networks  To facilitate the analysis, we have the following assumptions:

Control Algorithm Design
Assumption 1: The initial state of each WMR is the same in every iteration and , (0) (0) Assumption 2: The channel noise ( ), ( ) m t n t are bounded for every iteration, that is Assumption 3: Each switching communication graph has a spanning tree with a virtual leader as the root.
Remark 2： Assumption 1 is a basic condition of the ILC system, the identical initial condition is still followed in this paper.Furthermore, in [9], channel noise is assumed to be bounded, and the same assumption is still adopted in this paper.So, Assumption 1 and Assumption 2 are reasonable.

Figure 3
Block diagram of the scheme performed for WMR (7) by ( ), ( ) m t n t respectively, then the tracking error and the input data can be represented as then the consensus error can be rewritten as Remark 1: Due to the unreliability of the wireless network, the output errors , ( ) i k t δη received by the controller and input data , ( ) i k u t obtained by the actuator will be mixed with the channel noise during the transmission.

Figure 2
The WMR system transmits information over wireless networks  To facilitate the analysis, we have the following assumptions:

Control Algorithm Design
Assumption 1: The initial state of each WMR is the same in every iteration and , (0) (0) Assumption 2: The channel noise ( ), ( ) m t n t are bounded for every iteration, that is Assumption 3: Each switching communication graph has a spanning tree with a virtual leader as the root.
Remark 2： Assumption 1 is a basic condition of the ILC system, the identical initial condition is still followed in this paper.Furthermore, in [9], channel noise is assumed to be bounded, and the same assumption is still adopted in this paper.So, Assumption 1 and Assumption 2 are reasonable.

Figure 3
Block diagram of the scheme performed for WMR (8) then the consensus error can be rewritten as by ( ), ( ) m t n t respectively, then the tracking error and the input data can be represented as then the consensus error can be rewritten as Remark 1: Due to the unreliability of the wireless network, the output errors , ( ) i k t δη received by the controller and input data , ( ) i k u t obtained by the actuator will be mixed with the channel noise during the transmission.

Figure 2
The WMR system transmits information over wireless networks composed of the predicted learning term To facilitate the analysis, we have the following assumptions: Assumption 1: The initial state of each WMR is the same in every iteration and , (0) (0) Assumption 2: The channel noise ( ), ( ) m t n t are bounded for every iteration, that is Assumption 3: Each switching communication graph has a spanning tree with a virtual leader as the root.
Remark 2： Assumption 1 is a basic condition of the ILC system, the identical initial condition is still followed in this paper.Furthermore, in [9], channel noise is assumed to be bounded, and the same assumption is still adopted in this paper.So, Assumption 1 and Assumption 2 are reasonable.

Figure 3
Block diagram of the scheme performed for WMR (9) Remark 1: Due to the unreliability of the wireless network, the output errors , ( ) i k t δη received by the controller and input data , ( ) i k u t obtained by the actuator will be mixed with the channel noise during the transmission.

Control Algorithm Design and Convergence Analysis
To design a formation algorithm under iteration-varying communication topology, we design a switching signal function by ( ), ( ) m t n t respectively, then the tracking error and the input data can be represented as then the consensus error can be rewritten as Remark 1: Due to the unreliability of the wireless network, the output errors , ( ) i k t δη received by the controller and input data , ( ) i k u t obtained by the actuator will be mixed with the channel noise during the transmission.

Figure 2
The WMR system transmits information over wireless networks  Remark 2： Assumption 1 is a ba the ILC system, the identical ini still followed in this paper.Furt channel noise is assumed to be the same assumption is still a paper.So, Assumption 1 and As reasonable.

Control Algorithm Design and Convergence Analysis
With the above consensus error, the DILFC algorithm as shown in Fig. 3 is proposed: where 1 ( )

Control Algorithm Design and Convergence Analysis
With the above consensus error, the DILFC algorithm as shown in Fig. 3 is proposed: where 1 ( )

( ) α ⋅ h
For co the i th assump main re Theore assump (11) With the above consensus error, the DILFC algorithm as shown in Fig. 3 is proposed:

Control Algorithm Design and Convergence Analysis
With the above consensus error, the DILFC algorithm as shown in Fig. 3 is proposed:   ( ), ( ) m t n t respectively, then the tracking error and the input data can be represented as then the consensus error can be rewritten as Remark 1: Due to the unreliability of the wireless network, the output errors , ( ) i k t δη received by the controller and input data , ( ) i k u t obtained by the actuator will be mixed with the channel noise during the transmission.

Figure 2
The WMR system transmits information over wireless networks

Control Algorithm Design and Convergence Analysis
To design a formation algorithm under iteration-varying communication topology, we design a switching signal function to describe variable communication topology, where are identical in quantity and order.In the case that the topology structure changes with iteration, the model and consensus formation error are shown as follows With the above consensus error, the DILFC algorithm as shown in Fig. 3 is proposed: where 1 ( ) t Γ and 2 ( ) t Γ are bounded learning gain matrices.The control signal consists is composed of the predicted learning term To facilitate the analysis, we have the following assumptions: Assumption 1: The initial state of each WMR is the same in every iteration and , (0) (0) Assumption 2: The channel noise ( ), ( ) m t n t are bounded for every iteration, that is Assumption 3: Each switching communication graph has a spanning tree with a virtual leader as the root.
Remark 2： Assumption 1 is a basic condition of the ILC system, the identical initial condition is still followed in this paper.Furthermore, in [9], channel noise is assumed to be bounded, and the same assumption is still adopted in this paper.So, Assumption 1 and Assumption 2 are reasonable.
For convenience, we define , 1, 2, , i i N δ = … as the i th eigenvalue of L S + .Based on these assumptions, the following theorem gives the main results of this paper.
Theorem: Considering multiple WMRs satisfies assumptions 1~3, the DILFC controller is To facilitate the analysis, we have the following assumptions: Assumption 1: The initial state of each WMR is the same in every iteration and , (0) (0) Information Technology and Control 2021/3/50 Assumption 2: The channel noise ( ), ( ) m t n t are bounded for every iteration, that is Assumption 3: Each switching communication graph has a spanning tree with a virtual leader as the root.
Remark 2. Assumption 1 is a basic condition of the ILC system, the identical initial condition is still followed in this paper.Furthermore, in [9], channel noise is assumed to be bounded, and the same assumption is still adopted in this paper.So, Assumption 1 and Assumption 2 are reasonable.

Definition:
The α -norm is defined for a function For convenience, we define , … as the i th eigenvalue of L S + .Based on these assumptions, the following theorem gives the main results of this paper.
Theorem: Considering multiple WMRs satisfies assumptions 1~3, the DILFC controller is adopted shown in equation (12).If the gain matrix satisfies The α -norm of the system consensus error is bounded, and its upper bound is Proof: To simplify the analysis, the compact form of , , , ( ), ( ), ( ) and the channel noise are given adopted shown in equation (12).If the gain matrix satisfies The α -norm of the system consensus error is bounded, and its upper bound is Proof: To simplify the analysis, the compact form of , , , ( ), ( ), ( ) and the channel noise are given ( ) [ ( ) , ( ) , , ( ) [ ( ) , ( ) , , .
The multiple WMRs model can be written as Similarly, a compact form of consensus error can be given Taking norm on both sides of ( 16), we can get where Form ( 4) and ( 17), the input error could be expressed as ( ) Take the norm of both sides of the above The multiple WMRs model can be written as Similarly, a compact form of consensus error can be given where noise are given .
The multiple WMRs model can be written as Similarly, a compact form of consensus error can be given .
Then the system state and consensus error under channel noise are rewritten as where k represents change with iteration, substitute ( 18) into ( 12) .
According to (17) and (18), the input error could be expressed as where Then the system state and consensus error under channel noise are rewritten as noise are given .
The multiple WMRs model can be written as Similarly, a compact form of consensus error can be given where .
Then the system state and consensus error under channel noise are rewritten as Let k H L S = + ,where k represents change with iteration, substitute ( 18) into ( 12) .
According to (17) and (18), the input error could be expressed as where noise are given .
The multiple WMRs model can be written as Similarly, a compact form of consensus error can be given where .
Then the system state and consensus error under channel noise are rewritten as Let k H L S = + ,where k represents change with iteration, substitute ( 18) into ( 12) .
According to (17) and (18), the input error could be expressed as where Let k H L S = + , where k represents change with iteration, substitute (18) into (12) noise are given .
The multiple WMRs model can be written as Similarly, a compact form of consensus error can be given where .
Then the system state and consensus error under channel noise are rewritten as Let k H L S = + ,where k represents change with iteration, substitute ( 18) into ( 12) .
The multiple WMRs model can be written as Taking norm on both sides of ( 16), we can get where Form ( 4) and ( 17), the input error could be expressed as Taking norm on both sides of ( 16), we can get adopted shown in equation (12).If the gain matrix satisfies The α -norm of the system consensus error is bounded, and its upper bound is Proof: To simplify the analysis, the compact form of , , , ( ), ( ), ( ) and the channel noise are given ( ) [ ( ) , ( ) , , ( ) [ ( ) , ( ) , , .  The multiple WMRs model can be written as Similarly, a compact form of consensus error Taking norm on both sides of ( 16), we can get where Form ( 4) and ( 17), the input error could be expressed as ( ) where Form ( 4) and ( 17), the input error could be expressed as where (1 ) b ud z l b = + .
Based on Assumption 1, ( 23) can be rewritten as ( ) Substituting ( 24) into ( 21), it can be seen that Multiplying both sides of ( 25) by ( ) α to compute α -norm of the input error, where (22) Take the norm of both sides of the above equation and consider Property Form ( 4) and ( 17), the input error could be expressed as ( ) where (1 ) b ud z l b = + .
Based on Assumption 1, ( 23) can be rewritten as ( ) Substituting ( 24) into ( 21), it can be seen that Multiplying both sides of ( 25) by ( ) α to compute α -norm of the input error, where where Form ( 4) and ( 17), the input error could be expressed as ( ) where (1 ) b ud z l b = + .
Based on Assumption 1, ( 23) can be rewritten as ( ) Substituting ( 24) into ( 21), it can be seen that Multiplying both sides of ( 25) by ( ) Multiplying both sides of ( 25) by ( ) , we can have Since the norm of a constant is still a constant, (26) can be written as follows Through the above analysis, it is shown that the input converges gradually after multiple learning and iterations.To investigate the task target of formation tracking, this paper makes the following treatment: Similarly, multiplying both sides of ( 24) by (1 ) t α to compute α -norm of the state error, we get Since the norm of a constant is still a constant, ( 26) can be written as follows Since the norm of a constant is still a constant, ( 26) can be written as follows where Then, the ( 27) equals . There is always a matrix norm ensure ( ) where Since the norm of a constant is still a constant, ( 26) can be written as follows where Then, the ( 27) equals . There is always a matrix norm ensure ( ) Further, we can find α large enough so that ( ) 1 t ρ < .Then the upper bound of input error of mobile robot i can be input conv learning and target of for the followin Similarly, m (1 ) t α to co we get ( Then, the (27) equals Since the norm of a constant is still a constant, (26) can be written as follows where Then, the (27) equals . There is always a matrix norm ensure ( ) Define Since the norm of a constant is still a constant, (26) can be written as follows where Then, the (27) equals . There is always a matrix norm ensure ( ) .There is always a matrix norm ensure ( ) . Further, we can find α large enough so that ( ) 1 t ρ < .Then the upper bound of input error of mobile robot i can be expressed as max [1, ] z α > , we can have Since the norm of a constant is still a constant, (26) can be written as follows Through the above analysis, it is shown that the input converges gradually after multiple learning and iterations.To investigate the task target of formation tracking, this paper makes the following treatment: Similarly, multiplying both sides of ( 24) by (1 ) t α to compute α -norm of the state error, we get Through the above analysis, it is shown that the input converges gradually after multiple learning and iterations.To investigate the task target of formation tracking, this paper makes the following treatment: Information Technology and Control 2021/3/50 Similarly, multiplying both sides of ( 24) by (1 ) t α to compute α -norm of the state error, we get Through the above analysis, it is shown that the input converges gradually after multiple learning and iterations.To investigate the task target of formation tracking, this paper makes the following treatment: Similarly, multiplying both sides of ( 24) by (1 ) t α to compute α -norm of the state error, we get According to the above results, the formation error of the system is related to the channel noise ( ) k t m and ( ) k t n .Hence, when the channel noise is 0, lim || ( ) || 0 ∈ is a complex matrix, and the condition 1 ( ) ≤ < ∈ is specified since it includes all WMRs' kinematics and the complete communication topology.Next, we derive the convergence condition.For simplicity in the sequel, the time parameter is removed when no confusion arises.
A matrix norm operation [25] is defined: where U is a unitary matrix satisfying * ( ) U L S U Λ = + .Λ is an upper triangular matrix with diagonal entries being the eigenvalues of L S + , and * denotes matrix conjugate transpose.

{ , , }
Then, we can do the following calculation, According to the above results, the formation error of the system is related to the channel noise ( ) k t m and ( ) k t n .Hence, when the channel noise is 0, lim || ( ) || 0 ∈ is a complex matrix, and the condition is specified since it includes all WMRs' kinematics and the complete communication topology.Next, we derive the convergence condition.For simplicity in the sequel, the time parameter is removed when no confusion arises.
A matrix norm operation [25] is defined: pper n be input converges gradually after multiple learning and iterations.To investigate the task target of formation tracking, this paper makes the following treatment: Similarly, multiplying both sides of ( 24) by (1 ) t α to compute α -norm of the state error, we get ( ) According to the above results, the formation error of the system is related to the channel noise ( ) k t m and ( ) k t n .Hence, when the channel noise is 0, lim || ( ) || 0 is specified since it includes all WMRs' kinematics and the complete communication topology.Next, we derive the convergence condition.For simplicity in the sequel, the time parameter is removed when no confusion arises.
A matrix norm operation [25] is defined: where U is a unitary matrix satisfying * ( ) U L S U Λ = + .Λ is an upper triangular matrix with diagonal entries being the eigenvalues of L S + , and * denotes matrix conjugate transpose.

{ , , }
Then, we can do the following calculation, where U is a unitary matrix satisfying * ( ) . Λ is an upper triangular matrix with diagonal entries being the eigenvalues of L S + , and * denotes matrix conjugate transpose.
verges gradually after multiple d iterations.To investigate the task rmation tracking, this paper makes ng treatment: multiplying both sides of ( 24) by ompute α -norm of the state error, ). ( ) to the above results, the formation e system is related to the channel and ( ) k t n .Hence, when the channel is a complex matrix, and is specified ludes all WMRs' kinematics and the ommunication topology.Next, we e convergence condition.For in the sequel, the time parameter is hen no confusion arises.orm operation [25] is defined: is a unitary matrix satisfying ) S U .Λ is an upper triangular ith diagonal entries being the s of L S + , and * denotes matrix transpose.Then, we can do the following calculation, By the Assumption3, then, all the eigenvalues of k H have positive real parts.Thus, we can conclude that if By the Assumption3, then, all the eigenvalues of k H have positive real parts.Thus, we can conclude that if ( ) ( ( )) 1 This completes the proof.
By analyzing the above results, it can be found that the SC and CA channel noise will be accumulated along the iteration axis and time axis simultaneously.However, the summation of ( )  ( ) ( ( )) 1 This completes the proof.
By analyzing the above results, it can be found that the SC and CA channel noise will be accumulated along the iteration axis and time axis simultaneously.However, the summation of ( ) ∆ is far less than 1.Therefore, the accumulated channel noise in the iterative domain and time domain can be significantly suppressed by the sampling time.

Remark 3:
The formation control of a class of robots with repetitive motion is studied in this paper.In some practical applications, such as the mobile robot of logistics warehouse performing the throwing task and the agricultural robot performing the farm work task, which have strong repeatability.The analysis and design method in this paper can be directly extended.

Simulation
In this section, the formation tracking simulation of four WMRs by wireless network control is provided, and the communication topology changing with iteration is adopted.To verify the research effect of the previous section, we compared the influence of channel noise on the formation at different sampling times.
We consider the following reference trajectory: The desired trajectory deviation Information Technology and Control 2021/3/50 596 In addition, the initial state of each WMR satisfies assumption 1, that is, the initial state of each iteration is the same as that of the expected trajectory, which is set the following: Here are three communication diagrams for switching between WMRs, this is shown Fig. 4. It can be observed that only WMR 1 can receive the information of the leader, assuming that the information transmission between WMRs is directional.Then (1,0,0,0) S diag = , the Laplacian matrix can be known as: A randomly switching signal function ( ) k σ is defined and the value 1,2,3 respectively represents the corresponding topology.Fig. 5 shown the switching signal ( ) k σ in this simulation.The switching sequence takes the corresponding value as iteration changes.To ensure that the probability of each switching sig- By the Assumption3, then, all the eigenvalues of k H have positive real parts.Thus, we can conclude that if This completes the proof.
By analyzing the above results, it can be found that the SC and CA channel noise will be accumulated along the iteration axis and time axis simultaneously.However, the summation of ( ) ∆ is far less than 1.Therefore, the accumulated channel noise in the iterative domain and time domain can be significantly suppressed by the sampling time.
analysis and design method in this paper can be directly extended.

Simulation
In this section, the formation tracking simulation of four WMRs by wireless network control is provided, and the communication topology changing with iteration is adopted.To verify the research effect of the previous section, we compared the influence of channel noise on the formation at different sampling times.
We consider the following reference trajectory: The desired trajectory deviation Remark 3: The formation control of a class of robots with repetitive motion is studied in this paper.In some practical applications, such as the mobile robot of logistics warehouse performing the throwing task and the agricultural robot performing the farm work task, which have strong repeatability.The analysis and design method in this paper can be directly extended.The distributed algorithm ( 12) is adopted here, and the learning gain matrix And the matrix norm condition in the theorem is The Fig. 6 , respectively.It can be seen that with the increase of learning times, the formation effect is more and more perfect.At the same time, we also show the control signal in the 30 iterations through Fig. 7.
To illustrate that the proposed control algorithm has better transient performance in the time domain, the position and angular errors generated by the controller in reference [17] are compared.As shown in Fig. 8(a) and Fig. 8(b), it is obvious that the transient performance of the non-learning controller is not ideal in the And the matrix norm condition in the theorem is , respectively.It can be seen that with the increase of learning times, the formation effect is more and more perfect.At the same time, we also show the control signal in the 30 iterations through Fig. 7.
And the matrix norm condition in the theorem is , respectively.It can be seen that with the increase of learning times, the formation effect is more and more perfect.At the same time, we also show the control signal in the 30 iterations through Fig. 7. Next, we adjust different learning gains using controller (12) to compare their effects on control performance.The controller learning gain of the contrast group are selected as And the matrix norm condition in the theorem is The original gain is denoted as Γ , and the gain of the contrast group is denoted as ′ Γ .
The maximum position and angle errors of the  To illustrate the importance of the current learning item, a control algorithm (35) contains only a predicted learning item is adopted and the same simulation is performed again. , (   To illustrate the importance of the current learning item, a control algorithm (35) contains only a predicted learning item is adopted and the same simulation is performed again. , ( 1)) ( )( ( 1))}.To illustrate the importance of the current learning item, a control algorithm (35) contains only a predicted learning item is adopted and the same simulation is performed again.Its controller learning gain is the same as (12).
The maximum position and angle errors of the multiple WMRs under the controller ( 12) and (35) are shown in Fig. 9.It can be observed that the two control algorithms are different in the convergence speed and performance.The convergence result under controller (12) over the 30 iterations shows faster convergence speed and better transient performance.
Next, we adjust different learning gains using controller (12) to compare their effects on control performance.The controller learning gain of the contrast group are selected as  The original gain is denoted as Γ , and the gain of the contrast group is denoted as ′ Γ .
The maximum position and angle errors of the multiple WMRs under different learning gains are shown in Fig. 10.It can be observed that different learning gains affect the transient performance of the system, but there is no difference in the final convergence effect.

Figure 9
The influence of the current learning term on the maximum error of robot position and angle

Figure 10
The influence of different learning gains on the maximum error of robot position and heading f the ) and that n the of the contrast group is denoted as ′ Γ .
The maximum position and angle errors of the multiple WMRs under different learning gains are shown in Fig. 10.It can be observed that different learning gains affect the transient performance of the system, but there is no difference in the final convergence effect.

Figure 9
The influence of the current learning term on the maximum error of robot position and angle.

Figure 10
The influence of different learning gains on the maximum error of robot position and heading.
The tracking error convergence curve of the maximum position and angle of each WMR is shown in Fig. 11

Conclusion
In this paper, we studied the robust formation problem of multiple non-holonomic WMRs in repetitive motion environment, a distributed formation strategy based on ILC is proposed.The graph theory is introduced into the developed algorithm, which removes the condition that all WMRs need to obtain the desired trajectory information.The study used both predicted and current error as learning terms to improve the stability of the system The tracking error convergence curve of the maximum position and angle of each WMR is shown in Fig. 11(a) -Fig.11(d).Different sampling times were chosen for simulation, and the mean sum of CA and SC channel noises were 0.05 respectively.As can be seen from comparisons in these figures, the position and angle errors of WMR significantly decreased when a small sampling time was selected.
The tracking error convergence curve of the maximum position and angle of each WMR is shown in Fig. 11(a) -Fig.11(d).Different sampling times were chosen for simulation, and the mean sum of CA and SC channel noises were 0.05 respectively.As can be seen from comparisons in these figures, the position and angle errors of WMR significantly decreased when a small sampling time was selected.

Figure 11
The convergence of position and angle errors for all WMRs with different sampling times (a)

Co
In The tracking error convergence curve of the maximum position and angle of each WMR is shown in Fig. 11(a) -Fig.11(d).Different sampling times were chosen for simulation, and the mean sum of CA and SC channel noises were 0.05 respectively.As can be seen from comparisons in these figures, the position and angle errors of WMR significantly decreased when a small sampling time was selected.

Figure 11
The convergence of position and angle errors for all WMRs with different sampling times

Conclusion
In this paper, we studied the robust formation problem of multiple non-holonomic WMRs in repetitive motion environment, a distributed formation strategy based on ILC is proposed.

Conclusion
In this paper, we studied the robust formation problem of multiple non-holonomic WMRs in repetitive motion environment, a distributed formation strategy based on ILC is proposed.The graph theory is introduced into the developed algorithm, which removes the condition that all WMRs need to obtain the desired trajectory information.The study used both predicted and current error as learning terms to improve the stability of the system under channel noise and switching topologies.The sufficient conditions for the stability are derived theoretically.Through analysis, it is found that the norm of system tracking error is related to the function of system control input matrix and channel noise, and the accumulation of CA and SC noise in the time domain and iteration domain can be suppressed by reducing the sampling time.The results show that the proposed algorithm can realize the formation of multiple WMRs under channel noise and variable communication topology, and has better transient performance.At the same time, we observed that the number of times of learning is too much, which has a certain impact in practical engineering problems.This is also one of the questions that need further study in the future.

Figure 1
Figure 1 Configuration of WMR quantity and order.In the case that the topology structure changes with iteration, the model and consensus formation error are shown as follows , , , , composed of the predicted learning term of the system under the unpredictable uncertainty.
composed of the predicted learning term of the system under the unpredictable uncertainty.
of the system under the unpredictable uncertainty.
To design a formation algorithm under iteration-varying communication topology, we design a switching signal function ( ) : [0, ) {1, , M} k quantity and order.In the case that the topology structure changes with iteration, the model and consensus formation error are shown as follows , , , ,

Assumption 1 :Assumption 2 :
iteration and the current term Γ improve the stability of the sys unpredictable uncertainty.To facilitate the analysis, we hav assumptions: The initial state o the same in every iteration and x The Each switching graph has a spanning tree with as the root.

Figure 3
Figure 3Block diagram of the scheme perform quantity and order.In the case that the topology structure changes with iteration, the model and consensus formation error are shown as follows , , , , quantity and order.In the case that the topology structure changes with iteration, the model and consensus formation error are shown as follows , , , , stability of the system under the unpredictable uncertainty.

Figure 3
Figure 3 Block diagram of the scheme performed for WMR of the system under the unpredictable uncertainty.

Figure 3
Figure 3Block diagram of the scheme performed for WMR k Further, we can find α large enough so that ( ) 1 t ρ < .Then the upper bound of input error of mobile robot i can be Further, we can find α large enough so that ( ) 1 t ρ < .Then the upper bound of input error of mobile robot i can be Further, we can find α large enough so that ( ) 1 t ρ < .Then the upper bound of input error of mobile robot i can be is a constant matrix.
Figure 5 Switching signal among there different communication topologies

Figure 4 Communication topologies among WMRs Figure 5
Figure 4 Communication topologies among WMRs (a) and Fig. 6(b) show the formation effect of four WMRs at k = 5 and k = 30 when the channel noise ( Figure 6 Plane formation trajectory of WMRs

Figure 6 Plane
Figure 6Plane formation trajectory of WMRs

Figure 7 Control
Figure 7 Control signals for WMRs

Figure 6 Plane
Figure 6Plane formation trajectory of WMRs

Figure 7
Figure 7 Control signals for WMRs

[ 17 ]
are compared.As shown in Fig.8(a) and Fig.8(b), it is obvious that the transient performance of the non-learning controller is not ideal in the initial stage of control.By contrast, the learning controller can ensure the system has the ideal transient performance in the time domain after 30 iterations.

Figure 8
Figure 8Comparison of transient performance between learning controller and non-learning controller FigureThe in maxim

Figure 8
Figure 8Comparison of transient performance between learning controller and non-learning controller

Figure 8
FigureThe in maxim And the matrix norm condition in the theorem is

Figure 11
Figure 11The convergence of position and angle errors for all WMRs with different sampling times (a) -Fig.11(d).Different sampling times were chosen for simulation, and the mean sum of CA and SC channel noises were 0.05 respectively.As can be seen from comparisons in these figures, the position and In this paper, we studied the robust formation problem of multiple non-holonomic WMRs in repetitive motion environment, a distributed formation strategy based on ILC is proposed.The tracking error convergence curve of the , it can be seen that