In this paper, we mainly investigate the coordinated tracking control issues of multiple Euler–Lagrange systems considering constant communication delays and output constraints. Firstly, we devise a distributed observer to ensure that every agent can get the information of the virtual leader. In order to handle uncertain problems, the neural network technique is adopted to estimate the unknown dynamics. Then, we utilize an asymmetric barrier Lyapunov function in the control design to guarantee the output errors satisfy the time-varying output constraints. Two distributed adaptive coordinated control schemes are proposed to guarantee that the followers can track the leader accurately. The first scheme makes the tracking errors between followers and leader be uniformly ultimately bounded, and the second scheme further improves the tracking accuracy. Finally, we utilize a group of manipulator networks simulation experiments to verify the validity of the proposed distributed control laws.

With the rapid development of industrial technology, the industrial tasks are gradually becoming complicated and large-scale. When solving some complex industrial tasks, multiagent systems (MASs) gradually become the first choice due to its high reliability and economy, such as multiple robotic manipulator systems, spacecraft formation flying, and unmanned underwater vehicles [

So far, there mainly have been two consensus methods for the MASs coordinated control. The first control strategy is about the leaderless control method, which requires all state variables to gradually converge to a constant. For example, the authors used Lyapunov finite-time theory to propose a consensus control method to ensure the states converge to a constant based on undirected graphs in [

It is worth noting that all the above research studies about MASs are based on linear systems. However, linear systems have great limitations in MASs due to the existence of nonlinear uncertainties. Therefore, it is necessary to study the coordinated control problem of nonlinear systems [

Generally speaking, MASs are often affected by working environment, unknown dynamics, and unknown disturbances. The uncertainties will affect the work efficiency of MASs. To handle the uncertainties problem, the neural network (NN) technique is widely used for the MASs due to its good approximation ability [

In this study, the coordinated control problems for nonlinear multiple EL systems considering communication delays and output constraints are investigated. Compared with the existing papers, it uses a distributed observer to solve the communication delay problem of multiple EL systems. Then, the BLF technique is used to guarantee the time-varying output errors of the systems within the prescribed constraint boundary. We also utilize adaptive NNs to deal with the uncertain dynamics and the unknown disturbances of the multiple EL systems. The main contributions of the paper are summarized as follows.

Considering communication delays among different followers, the input signal source is regarded as a virtual leader. In addition, a distributed observer is used to ensure that the virtual leader’s state information can be obtained by all followers.

Two distributed control schemes are designed to guarantee the tracking errors are UUB and asymptotically converge to origin.

The adaptive NN technique is utilized to compensate the uncertain dynamics of the multiple EL systems.

Based on the BLFs, an asymmetric BLF is designed to make the output errors satisfy the time-varying output constraint requirements.

In the following research, Section

The basic mathematical symbols and definitions in this paper are shown in Table

Mathematical symbols in this study.

Mathematical symbol | Definition |
---|---|

The set of real numbers | |

The set of | |

The set of | |

Diagonal matrices whose diagonal elements are | |

Kronecker product | |

The minimum eigenvalue of a real symmetric matrix | |

The maximum eigenvalue of a real symmetric matrix | |

Matrix trace |

The communication interactions among a virtual leader and

A directed spanning tree is contained within

We define the element

The multiple EL systems contain

The disturbance

The following properties of EL systems (

The matrix

We assume that

When all followers can obtain

If the coefficient matrices

The useful lemmas used in this study are shown as follows.

(see [

(see [

(see [

(see [

In this study, if the trajectory tracking errors are too large, it may cause undesired losses. Therefore, the output states of the followers should be limited. If the time-varying bounds are

When considering the influence of tracking errors on multiple EL systems, a time-varying BLF is used in this study to ensure that the output

First, an auxiliary variable is defined as

Then, we define the following error variables:

Inspired by He et al. [

Consider that asymmetric BLF is an improvement of symmetric BLF, which can better adapt to the requirements of time-varying output constraints and ensure the high trajectory tracking accuracy for multiple EL systems. An asymmetric BLF is chosen as

The output tracking error variables are transformed as follows:

Substituting (

From (

The virtual control

When only part of the followers can get the state information, we use the following distributed observer [

(see [

Since

According to [

For the EL system (

Differentiating (

Substituting (

In this study,

The distributed adaptive control schemes are designed as (

Since

Because

The matrices have the following properties:

Further we have

Substituting (

Thus, (

We can know

We can ensure

It can be seen from (

Moreover,

By substituting (

From (

According to (

From (

From (

Equation (

According to [

For the EL system (

By introducing the continuous function

For the EL system (

When

Select the same

Similar to (

For a small positive constant

Since

Moreover, we have

Due to the fact that

Therefore, we have

From (

Following the similar steps as (

The

When

If

According to (

The results of this paper can provide some reference for formation control of MASs. In the future, the formation control problems for MASs considering time-varying communication delays and full-state constraints will be studied.

In this section, we utilize some examples to verify the validity of the proposed schemes in practical aspect. In this study, we consider 4 2-degree-of-freedom robotic manipulators as an example for the simulation experiment. The communication topology is shown in Figure

Communication topology.

The structure of robotic manipulator.

The dynamic equation of the

In this section,

Table

The parameters of the robotic manipulators.

Parameter | Manipulator 1 | Manipulator 2 | Manipulator 3 | Manipulator 4 |
---|---|---|---|---|

1.01 | 0.96 | 1 | 1.04 | |

1.12 | 1.14 | 1.03 | 1.09 | |

1 | 0.95 | 0.98 | 1 | |

0.96 | 1 | 0.95 | 1 | |

0.23 | 0.21 | 0.19 | 0.21 | |

0.41 | 0.40 | 0.42 | 0.41 |

We set the time-varying output constraints as follows:

The boundary value of tracking error

Thus, we have

The initial angle of robotic manipulators is set as follows.

For the

We select Gauss function as the activation function, and its form is

The desired trajectory of the virtual leader is designed as follows:

The state variable

In practical physical systems, we often need to limit the amplitude of the control inputs

In this section, we choose the communication delay as

For controller 1, we choose the parameters as

The generalized coordinates

The generalized coordinates

The auxiliary variables

The auxiliary variables

The control input torque

The control input torque

Constraint effect of

Constraint effect of

Constraint effect of

Constraint effect of

Figures

For the improved algorithm by using discontinuous sign function on Algorithm 1, let

The control input torque

The control input torque

For Algorithm 2, we set

The generalized coordinates

The generalized coordinates

The auxiliary variables

The auxiliary variables

The control input torque

The control input torque

Constraint effect of

Constraint effect of

Constraint effect of

Constraint effect of

From Figures

Based on the above simulation results, we can discover that all followers have very good tracking effect on the leader, and the tracking error always satisfies the time-varying constraints. In addition, the tracking errors

In this study, we propose two practical control strategies to address distributed coordinated tracking problem for the multiple EL systems subjected to communication delays and time-varying constraints. The distributed observer is used to cope with the communication delays for multiple EL systems. We utilize the NN technique to compensate nonlinear uncertainties. At the same time, an asymmetric BLF is used to guarantee that the output errors are always within the output constraints. The adaptive control Algorithm 1 is designed to ensure that the tracking errors is designed to ensure that the tracking errors among the followers and the virtual leader can be bounded. Based on Algorithm 1, the improved Algorithm 2 can make the tracking errors smaller. The simulation results indicate that the proposed methods can effectively solve the problem of communication delays and make the tracking errors meet the prescribed output constraints.

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.

This study was supported by the National Natural Science Foundation of China (grant nos. 61803119, U1713205, and 51779058).