This research presents an adaptive energy-saving _{2} closed-form control approach to solve the nonlinear trajectory tracking problem of autonomous mobile robots (AMRs). The main contributions of this proposed design are as follows: closed-form approach, simple structure of the control law, easy implementation, and energy savings through trajectory tracking design of the controlled AMRs. It is difficult to mathematically obtained this adaptive _{2} closed-form solution of AMRs. Therefore, through a series of mathematical analyses of the trajectory tracking error dynamics of the controlled AMRs, the trajectory tracking problem of AMRs can be transformed directly into a solvable problem, and an adaptive nonlinear optimal controller, which has an extremely simple form and energy-saving properties, can be found. Finally, two test trajectories, namely circular and S-shaped reference trajectories, are adopted to verify the control performance of the proposed adaptive _{2} closed-form control approach with respect to an investigated _{2} closed-form control design.

_{2}closed-form control

In recent decades, comprehensive applications of autonomous mobile robots (AMRs) have attracted considerable attention. These AMRs with extended energy endurance, more precise motion ability, and effective control approaches have been applied in the transportation, security, and inspection domains. Thus, a precise motion controller for AMRs and energy saving are becoming increasingly important in the robotics application field, which have been discussed in many studies [_{2} [

For these reasons, an innovative nonlinear energy-saving control approach with a simple control structure that can provide high-performance trajectory tracking for AMRs is presented in this paper. To reduce computational costs and output low-energy torque, a novel energy-saving adaptive _{2} closed-form control approach for the trajectory tracking of AMRs is developed. Furthermore, this problem is directly solved using a nonlinear time-varying differential equation. Moreover, the proposed adaptive _{2} closed-form solution must satisfy an _{2} optimal performance index. In such a circumstance, it is extremely difficult to obtain the solution of a nonlinear time-varying differential equation. However, this solution can be expanded and inferred by selecting suitable state variable transformations and performing mathematical analyses of the dynamic equations of the trajectory tracking error. With such a solution, the adaptive _{2} closed-form control approach for the trajectory tracking of AMRs will have a direct implementation structure and provide energy saving.

The remainder of this paper is organized as follows. Section 2 describes the mathematical model of trajectory tracking error of AMRs. In Section 3, the adaptive _{2} closed-form controller design for AMR trajectory tracking is described. Section 4 illustrates the simulation results obtained for AMRs by using the proposed approach. Finally, our concluding remarks are given in Section 5.

The trajectory tracking error mathematical model of AMRs is presented in this section. Based on the standard trajectory tracking error mathematical equation and the geometry relationship between the AMR and global coordinate systems, a controlled AMR with a nonlinear trajectory tracking error dynamic equation can be inferred as follows.

In _{c}, _{c}}. In addition,

Under the nonslipping condition, a standard AMR system usually moves along the orientation of the driving wheels’ axis. Hence, the kinematics of the controlled AMR with constraints can be expressed using the following equation [

where

In this study, the dynamics of the controlled AMR are inferred using the Euler–Lagrange method, as expressed in

where

Details of the AMR dynamics are as follows:

where

Suppose

where

According to

Mathematically, it is difficult to use

where

where

From

where

and

If

then the dynamic equation of trajectory tracking error can be revised as follows:

where

An analytic adaptive _{2} control law for the AMR is deduced from the following equations. To inspect _{2} controller design of trajectory tracking with the _{2} performance property of AMRs can be solved if there exists a closed-form solution

The aforementioned performance index can be achieved for all

In this section, we solve the AMR trajectory tracking control problem described in Section 2. To this end, we present a novel energy-saving adaptive _{2} closed-form control approach for trajectory tracking of the AMR based on the following nonlinear adaptive _{2} closed-form control theorem.

_{2} closed-form control law

where

and

If _{2} time-varying differential _{2} trajectory tracking problem of the AMR in

It is difficult to determine the adaptive _{2} closed-form solution and solve the nonlinear time-varying differential equations in _{2} closed-form solution can be directly derived from

In general, it is difficult to solve

Because state-space transformation matrix

where

To investigate the second and third terms on the left-hand side of time-varying differential

By using the results of

In addition, the optimal control law and adaptive law can be expressed as

where

where

Using the definitions of

with

Then, the following adaptive _{2} control law can be used to solve the trajectory tracking problem of adaptive _{2} closed-form control.

where

In this section, a verification scenario with the _{2} closed-form and adaptive _{2} closed-form control approach for trajectory tracking of a circle and an S shape is presented using the MATLAB software application. According to the aforementioned simulation results, this adaptive _{2} closed-form control approach will be certified the performances of trajectory tracking and energy saving of the AMR are more excellent than _{2} closed-form control approach.

To construct the simulation environment, the following parameters of the practical AMR are employed:

where

_{2} closed-form and adaptive _{2} closed-form control approaches for tracking a desired circular trajectory with a radius of 3.8 m. The verification results of the _{2} closed-form and adaptive _{2} closed-form control approaches for tracking this circular trajectory are displayed in _{2} closed-form and adaptive _{2} closed-form control approaches for the desired circular trajectory is adequate. The circular trajectory tracking errors along the x-y axis and angle-to-convergence rates obtained using the _{2} closed-form and adaptive _{2} closed-form control approach are displayed in _{2} closed-form control approach can track the desired circular trajectory of the AMR more rapidly and yield superior trajectory tracking performance and energy savings than the _{2} closed-form control approach.

In the second simulation scenario, the verification results of the _{2} closed-form and adaptive _{2} closed-form control approaches for an S-shaped trajectory with a radius of 3.8 m are displayed in _{2} closed-form control approach for the desired S-shaped trajectory illustrate an outstanding performance, as displayed in _{2} closed-form control approach can track the desired S-shaped trajectory faster than the _{2} closed-form control approach, and the energy-saving effect of the adaptive _{2} closed-form control approach is superior to that of the _{2} closed-form control design in this simulation scenario.

Suboptimal trajectory tracking designs have been studied for autonomous mobile wheel robots in the past decades, and most of them have achieved acceptable control performance. However, they have disadvantages such as their extremely complex control structures, such as the sliding mode and backstepping control methods. For simultaneously achieving satisfactory tracking performance and a simple control structure, an analytical adaptive nonlinear control scheme was developed to track the trajectory of autonomous mobile wheel robots in this study. The proposed adaptive control design consists of an adaptive cancellation term that is used to cancel the nonlinear component of tracking errors and an optimal control term to minimize the power consumption when tracking the desired trajectories. Thus, the proposed control method has an impressive property; that is, without knowing the system parameters of autonomous mobile wheel robots, the desired trajectory tracking performance can be maintained by exploiting the adaptive learning ability of the proposed method. The simulation results indicate that the proposed adaptive nonlinear control method delivers promising trajectory tracking performance for WMRs because the tracking errors quickly converge to zero when a large amount of modeling uncertainties appear. Therefore, the proposed method has the advantages of being able to execute tasks such as the uploading and downloading of goods and regular patrolling.

_{2}nonlinear control approach

_{2}and H

_{∞}performance objectives I: Robust performance analysis