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In this paper, a robust adaptive output feedback control strategy based on a sliding mode super-twisting algorithm is designed for the trajectory tracking control of a wheeled mobile robot. First, a robust adaptive law is designed to eliminate the influence of parameter uncertainties. Second, a double-power sliding mode surface is designed to improve the response speed of the robot system. Finally, a high-gain observer is employed to estimate the speed information. The stability of the closed-loop system is proved using the Lyapunov stability theorem. The effectiveness of the proposed control strategy is verified by simulation.

As a kind of practical operating tools, mobile robots have far-reaching application prospects in various industries. Mobile robots are highly nonlinear, continuous time-varying, strongly coupled dynamic systems. Therefore, it is an extremely challenging task to realize that mobile robots can overcome different external disturbances and track a predetermined trajectory in dissimilar environments. The parameters of their motion control system must be adjusted with the changes of the tracking path and the surrounding environment at all times to meet the requirements of precise control.

Nevertheless, through the unremitting efforts of researchers, the robust adaptive control method [

Although the traditional PID control has the characteristics of a simple principle, few parameters, and easiness of use, it has been extensively used in industrial process control. However, it is a linear combination of the three items of error proportional, integral, and derivative action, and it does not possess the function of adjusting gain parameters online. Therefore, traditional PID control is very limited to the tracking control of the complex nonlinear system signals such as mobile robots. With in-depth research of mobile robots by researchers, sliding mode control (SMC) technology has become a mature trajectory tracking control method for mobile robots. Sliding mode control has a lot of advantages, such as fast response and strong robustness to disturbances and system uncertainties. However, the main problem with this sliding mode method is the inevitable chattering phenomenon, which may harm the actuator. There are numerous techniques to reduce chattering, such as using the saturation function or the sigmoid function instead of the discrete control function [

To achieve the purpose of output feedback for the control system, the observer is employed in mobile robots. Ovalle et al. [^{th}-order perturbed integral system is introduced in [

It is worth noting that a controller based on output feedback control is not common in mobile robot systems. It is also noteworthy that, in the study of mobile robot control strategies, most of the literature is based on three-wheel mobile robots. However, driving in real life, using four wheels is more conducive to keep the balance and stability of the robot. Therefore, this paper is intended to design the controller to achieve precise control of the four-omnidirectional-wheel mobile robots. To some extent, this article is inspired by study in [

In practical application, the use of a speed sensor to obtain the speed signal of the mobile robot makes its measurement accuracy vulnerable to the interference of climate and environment. Therefore, to prevent the use of high-cost speed sensors, a high-gain observer is designed to estimate the speed signal of the autonomous mobile robot effectively in real time, and the output feedback control of the system is achieved.

Double-power sliding mode surface is designed to effectively speed up the convergence of the system.

Uncertain parameters are considered, and a robust adaptive law that improves the robustness and trajectory tracking accuracy of the system is intended to fit the uncertain parameter.

The resulting trajectory tracking error is very small, and the error level is approximately

The structure of the article is as follows: the model description of the mobile robots is introduced in Section

In this paper, the omnidirectional mobile robot driven by four wheels is taken as the research object. The driving wheel adopted is the Mecanum wheel. The Mecanum wheel is compact and flexible, and it is a very lucrative omnidirectional wheel. It has been extensively employed in engineering applications. When the wheel rotates in a 45° arrangement, the free roller is connected to the ground, and the ground will create a 45° friction force between the wheel and the rotating shaft clip. The friction force can be split into the

Structure diagram of the mobile robot.

The dynamic equation of the omnidirectional mobile robot is described in [

We represent model (

For the sake of convenience,

In practice, it is difficult to directly measure the speed signal of a mobile robot owing to complex environmental interference. To avoid the use of high-cost velocity sensors, this paper uses separate principles to design the observer and controller separately. This chapter designs a high-gain observer for real-time estimation of the speed signal of the mobile robot during operation and realizes the output feedback control of the system.

The high-gain observer is designed as follows:

Let

For the convenience of expression, we define the following symbolic rules:

For any choice of

For the particular case in which

Let

There is a known normal number

We define the observer estimation error of velocity state quantity as

If the observer given in (

The proofs of all theorems will be provided in Appendix.

As we all know, the trajectory tracking problem of mobile robots can be regarded as the problem of target convergence. That is tantamount to designing a suitable control strategy so that the mobile robot gradually approaches the target point driven by the control output. For this reason, this section will design an improved adaptive super-twisting sliding mode trajectory tracking controller for mobile robots.

Assume that the desired position is

In order to design a super-twisting algorithm to solve the trajectory tracking problem, the sliding mode surface is defined as follows:

Since the speed information of the mobile robot is difficult to measure directly, the estimated value of the high-gain observer is used instead. Therefore,

Differentiation of formula (

In the design of sliding mode control system, it is necessary to design an ideal equivalent control law

Simultaneous equations (

To solve the chattering problem in sliding mode control, the discontinuous term

However, the disturbance term

To ensure system stability, implying that

Now, to make the mobile robot system have faster convergence speed and better motion quality. It is necessary to improve the sliding mode surface (

In addition, it is worth noting that the mobile robot is susceptible to external interference during actual operation, which affects the trajectory tracking task. This requires that the control parameters of the mobile robot must be adjusted online in real time with the changes in the environment to make the mobile robot adaptive.

In the sliding mode surface given in equation (

The designed robust adaptive control law is given by

Consequently, equation (

Finally, the input torque can be converted into the input voltage of the motor by the following formula:

Based on Assumption

To demonstrate the advantages of the improved algorithm, the simulation results are assessed. In this paper, four simulation comparisons are carried out with the same mobile robot model. The methods being compared are the conventional PID control, the super-twisting sliding mode high-order observer control method (STSM-HOSMO) given in [

Block diagram of the super-twisting control based on a high-gain observer.

The related parameters are given in Table

Related parameters.

Symbol | Value |
---|---|

2.42 | |

7.19 | |

1.7 | |

10.22 | |

1.42 | |

51.11 | |

2 | |

1 | |

0.05 | |

0.6 | |

0.0001 | |

0.001 | |

100 | |

7 kg | |

0.4 kg | |

64 | |

1.9 | |

0.013 Nm/A | |

0.0133 Vs/rad | |

0.001 | |

0.05 m | |

0.22 m | |

0.35 m | |

The PID controller is given by

The gains are given as

The desired tracking trajectory of the set mobile robot is

The disturbance items are set to

The simulation results are as follows.

In the simulation verification, to verify the effectiveness of the designed control algorithm for different tracking paths, this paper traces three kinds of trajectories: respectively, circular trajectory, eight-shaped trajectory, and anti-C shaped trajectory. The tracking response is illustrated in Figures

Circular reference trajectory tracking response diagram.

Eight-shaped reference trajectory tracking response diagram.

Anti-C-shaped reference trajectory tracking response diagram.

In the following, a further analysis of the circular reference trajectory will be carried out. The simulation results in Figure

Trajectory tracking error.

Figure

The sliding surface of [

The output of the robust adaptive strategy.

Comparison chart of two observers, respectively, estimating pose signals.

Temporal evolution of the control inputs.

To further verify the robustness of the control strategy designed in this paper, in 20 seconds, we suddenly added a 50 kg load disturbance to the mobile robot system, and the simulation results are shown in Figures

Circular reference trajectory tracking response diagram with disturbance.

Trajectory tracking error with disturbance.

The output of the robust adaptive strategy with disturbance.

It can be observed in Figures

In order to address the trajectory tracking problem of a class of mobile robots driven independently by four omnidirectional wheels, a robust adaptive super-twisting sliding mode controller is proposed, and the motion control problem of parameter uncertainty and external disturbance is solved. In this paper, a double-power sliding mode surface is designed for the super-twisting sliding mode controller, which accelerates the response speed of the robot system, and the system uncertainty is better estimated by the addition of the robust adaptive controller, which makes the system more robust to uncertain factors such as external disturbances. By comparing the high-gain observer with the high-order observer, we find that the high-gain observer used throughout this paper makes the trajectory tracking control have better stability and accuracy in the implementation of output feedback control. The stability of the system is demonstrated by using the Lyapunov theory. The simulation results also verify the effectiveness of the proposed control strategy.

Define the scalar estimation error as follows:

Then,

Because

By formula (

Therefore, we choose the Lyapunov function as the solution of the equation

Choose

The following conclusion is drawn in accordance with Theorem 4.5 in the book

Therefore,

Choose a positive candidate Lyapunov function as

The derivation of (

Substituting equation (

Since

Sinc

Then, according to the generalization of the standard Lyapunov’s stability theorem, the closed-loop system is uniformly ultimately bounded. This completes the proof.

Data are not made available as the funding agency does not allow data sharing of the experimental program and this involves intellectual property issues.

The authors declare that they have no conflicts of interest.

This work was supported by the Natural Science Foundation of Guangdong Province in China (Grant no. 2018A0303130076), the Science and Technology Planning Project of Zhanjiang City (Grant no. 2017A02025), the Fostering Plan for Major Scientific Research Projects of the Education Department of Guangdong Province-Characteristic Innovation Projects (Grant no. 2017KTSCX087), the 2019 “Chong First-Class” Provincial Financial Special Funds Construction Project (Grant no. 231419019), and the Guangdong Science and Technology Project (Grant no. 2013B011304009).