Development of an Adaptive Trajectory Tracking Control of Wheeled Mobile Robot

Classical modeling and control methods applied to differential locomotion mobile robots generate mathematical equations that approximate the dynamics of the system and work relatively well when the system is linear in a specific range. However, they may have low accuracy when there are many variations of the dynamics over time or disturbances occur. To solve this problem, we used a recursive least squares (RLS) method that uses a discrete-time structure first-order autoregressive model with exogenous variable (ARX). We design and modify PID adaptive self-adjusting controllers in phase margin and pole allocation. The main contribution of this methodology is that it allows the permanent and online update of the robot model and the parameters of the adaptive self-adjusting PID controllers. In addition, a Lyapunov stability analysis technique was implemented for path and trajectory tracking control, this makes the errors generated in the positioning and orientation of the robot when performing a given task tend asymptotically to zero. The performance of the PID adaptive self-adjusting controllers is measured through the implementation of the criteria of the integral of the error, which allows to 1 Instituto Tecnológico Metropolitano (Medellín-Antioquia, Colombia). guiovannysuarez97899@correo.itm.edu.co. ORCID: 0000-0003-4131-9280 2 M. Sc. Politécnico Colombiano Jaime Isaza Cadavid (Medellín-Antioquia, Colombia). ndmunoz@elpoli.edu.co. ORCID: 0000-0002-4696-8151 3 M Sc. Instituto Tecnológico Metropolitano (Medellín-Antioquia, Colombia). henryvasquez@itm.edu.co. ORCID: 0000-0001-5428-0253 Development of an Adaptive Trajectory Tracking Control of Wheeled Mobile Robot Revista Facultad de Ingeniería (Rev. Fac. Ing.) Vol. 30 (55), e12022. Enero-Marzo 2021. Tunja-Boyacá, Colombia. L-ISSN: 0121-1129, e-ISSN: 2357-5328. DOI: https://doi.org/10.19053/01211129.v30.n55.2021.12022 determine the controller of best performance, being in this case, the PID adaptive self-adjusting type in pole assignment, allowing the mobile robot greater precision in tracking the trajectories and paths assigned, as well as less mechanical and energy wear, due to its smooth and precise movements.

However, they may have low accuracy when there are many variations of the dynamics over time or disturbances occur. To solve this problem, we used a recursive least squares (RLS) method that uses a discrete-time structure first-order autoregressive model with exogenous variable (ARX). We design and modify PID adaptive self-adjusting controllers in phase margin and pole allocation. The main contribution of this methodology is that it allows the permanent and online update of the robot model and the parameters of the adaptive self-adjusting PID controllers.
In addition, a Lyapunov stability analysis technique was implemented for path and trajectory tracking control, this makes the errors generated in the positioning and orientation of the robot when performing a given task tend asymptotically to zero.
The performance of the PID adaptive self-adjusting controllers is measured through the implementation of the criteria of the integral of the error, which allows to determine the controller of best performance, being in this case, the PID adaptive self-adjusting type in pole assignment, allowing the mobile robot greater precision in tracking the trajectories and paths assigned, as well as less mechanical and energy wear, due to its smooth and precise movements.

I. INTRODUCTION
Generally, studies on trajectory tracking of mobile robots are based on classical mathematical models described in the literature, which represent in a very simplified way the kinematics of the robot, such as [1,2]. In some cases, mathematical equations are included that roughly represent the dynamics of the system [3]. These equations work relatively well when the system is linear in a specific range, as proposed by Alves [4], the mathematical model of the motors of the traction system is only obtained from physical laws, so the model may present low accuracy when there are many variations of the system dynamics over time, uncertainty or disturbances.
Related works try to solve this problem in different ways, some authors focus on analyzing the impact of parametric uncertainties of a kinematic model on the estimation of the speed and pose of the robot, which provides important information for the design of the controllers [5]. Abdelwahab [6] presents a rule-based heuristic control system with fuzzy logic, which has proven to be useful in addressing uncertainty and imprecision to achieve robust and smooth trajectory tracking.
Ortigoza [7] also proposes as a strategy to develop models and controllers for each subsystem that makes up the robot, including the power stage. Dobribarsci [8] proposes the identification of motors up to the design of linear and non-linear controllers. In these and other articles it is common that the models obtained do not include updates over time.
The objective of the research is to present a methodology for the identification, modeling and optimal control of a mobile robot in trajectory tracking tasks [9], for which online identification is performed, that is, the model is being updated in real time, constituting a significant contribution to making the model and robot controllers more accurate. A discrete-time parametric model is used, among which the autoregressive model with exogenous variable ARX [10,11] stands out.
This article is structured as follows: section 2 describes the hardware, software and methodology used for identification, mathematical modeling and control, section 3 presents the results and analysis, section 4 presents the conclusions.

II. MATERIALS AND METHODS
This section describes the hardware and software materials, and the methodology for identifying, modeling, and controlling the robot for trajectory tracking missions.

A. Mobile Robot
With the Lego Mindstorms EV3 [12] educational kit, a differential locomotion mobile robot was built, as shown in figure 1.

B. Software
The following software was used in the development of the project: 1. Operating system Microsoft Windows 10 professional.

1) Direct Kinematics of the Mobile Robot.
To obtain the kinematic behavior data of the differential robot, it is assumed that the robot moves on a flat surface without friction, that it moves only by the rotational motion of the wheels, that it is considered as a solid, rigid mechanism without flexible parts, but that the holonomic constraints of the system must be considered [13][14], that is, it cannot move to the sides, as  The measurements of interest in the modeling process are the distance between the wheels, called L and the radius of the wheels, called R, as shown in figure 3. To achieve precise movement, control must be exercised over the right and left wheel velocities, VR and VL, which affect the states of the system x, y, Ɵ. The robot's linear and angular velocities are defined in equations (1) and (2), from which equations (3), (4) and (5) define the kinematics of the robot's motion on each axis.

2) Inverse Kinematics of the Mobile Robot.
To control the movement of the robot, the angular velocities of the robot wheels are altered, which are determined from the desired linear and angular velocities, which are represented by equation (9).   In this representation, the position error vector is given by equation (12).
This error corresponds to the robot's reference system {X R , Y R , Ɵ}, where Y Ĝ , X Ĝ , are the coordinates of the target. If α ϵ I, where: Applying transformation to polar coordinates considering as its origin the objective point, we obtain: Where ƿ is the distance between the moving robot and the target, α is the angle needed to orient the robot towards the target point. Finally, β is the angle of orientation of the robot with respect to the coordinate system of the target point.
The task of controller design is to find a control matrix, which is given by equation (17), in equation (18)     (21) con 1 > 0 = 2 * α + 1 * cos(α) * sin(α) * (α + 2 * β)/α (22) con 2 > 0 Figure 8 shows the forward and inverse kinematics of the differential drive robot. where the robot starts from position (0,0), and 0° orientation degrees, it is observed that the robot requires more time to reach the desired trajectory, the opposite happens for the straight-line trajectory, it influences the computational expenditure by the processing system and energy consumption by the traction system. Table   1 shows the criterias of the error integral [15] used for selecting the PID adaptive controller. the controller that has the best results is the self-adjusting adaptive in pole assignment.

IV. CONCLUSIONS
The methodology of identification, modeling and control presented in this article allows the permanent online update of the locomotion system engine models, as well as the parameters of the controllers designed to track the trajectories of the differential drive robot, which is an advantage because it achieves more accurate system response compared to traditional identification, modeling and control techniques that have a limited range of operation.
The online identification method implemented allows knowing at any instant of time the mathematical model of the motors, an aspect that is not possible in those techniques where the model is only obtained from physical laws.
The online parameters update of the trajectory tracking controllers, allows movements with precision and smoothness, compared to results obtained with the classic PID controllers evaluated. The execution of smooth trajectories is associated with less control effort and lower energy consumption [16].
The implementation of controllers from the Lyapunov stability analysis, facilitates to the system to work optimally in the event of unexpected changes or disturbances, consequently, the robot may have a better cost-benefit ratio in energy consumption, smoothness of the trajectory, precision of movements, etc.
The proposed methodology provides a strategy to solve path and trajectory tracking control missions. Simulations and experiments were performed in real environments to verify their robustness and efficiency. In addition, taking mathematical models to first-order systems favors the low computational cost of the proposed solution.