Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 608015, 15 pages

http://dx.doi.org/10.1155/2015/608015

## Motion Control Design for an Omnidirectional Mobile Robot Subject to Velocity Constraints

^{1}Departamento de Ingeniería Eléctrica y Electrónica, Instituto Tecnológico de Sonora, Campus Náinari, Antonio Caso 2266, Villa ITSON, 85130 Ciudad Obregón, SON, Mexico^{2}Departamento de Electrónica y Telecomunicaciones, CICESE, Carretera Ensenada-Tijuana No. 3918, Zona Playitas, 22860 Ensenada, BC, Mexico^{3}Departamento de Posgrado, CIDETEC, Instituto Politécnico Nacional, Sección de Mecatrónica, Juan de Dios Bátiz s/n, Nueva Industrial Vallejo, Gustavo A. Madero, 07700 México, DF, Mexico^{4}Universidad Autónoma del Carmen, Facultad de Ingeniería, Calle 56 No. 4, Benito Juárez, 24180 Ciudad del Carmen, CM, Mexico

Received 31 October 2014; Revised 8 February 2015; Accepted 15 February 2015

Academic Editor: Nazim I. Mahmudov

Copyright © 2015 Ollin Peñaloza-Mejía et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A solution to achieve global asymptotic tracking with bounded velocities in an omnidirectional mobile robot is proposed in this paper. It is motivated by the need of having a useful in-practice motion control scheme, which takes into account the physical limits of the velocities. To this end, a passive nonlinear controller is designed and combined with a tracking controller in a negative feedback connection structure. By using Lyapunov theory and passivity tools, global asymptotic tracking with desired bounded velocities is proved. Simulations and experimental results are provided to show the effectiveness of the proposal.

#### 1. Introduction

Control of mobile robots has received great attention in last decades. The interest is motivated by the different potential applications in which these systems are very useful (e.g., indoor/outdoor navigation in wide areas). The fundamental control problems addressed have been regulation and trajectory tracking, and several control techniques have been proposed to successfully solve them (see, e.g., [1]). Also, the demands for mobile robots with high mobility have increased in the last years. Since the conventional mobile robot does not have high mobility due to its nonholonomic constraint, the omnidirectional mobile robot (OMR) has attracted the attention, because of its ability to move simultaneously, and independently in an arbitrary direction in the horizontal plane (translational motion) and with different orientation (rotational motion). Hence, the OMR is a kind of holonomic robot [2]. Because of its Euler-Lagrange dynamics, the recognized “classical” control techniques for robot manipulators [3] have been adapted for controlling this kind of mobile robot. Recently, intensive and notable research has been devoted to this device (see, e.g., [1, 4–6] and references therein).

To implement the control strategies, a localization system is required to estimate the actual wheeled mobile robot positions and velocities [7], which are mainly based on the odometric data, the robot kinematics, and the Kalman filter. Nevertheless, since there exists a maximum sampling time to reconstruct the velocity of the wheels, the odometric data could have errors if the Nyquist-Shannon criteria are not satisfied. Hence, the important issue to consider during the control design is the inclusion of the physical limitations of the system, such as bounds in velocities and/or torques. For instance, high velocities developed by the mobile robot can result in a higher probability of wheel slides and deviations from the desired trajectory. In practice, the mobile robot has bounded velocities and torques which should never be exceeded. If the control strategy is designed without considering these constraints, the system could exhibit poor performance and even instability [8].

Many control strategies in robotics have been proposed without considering explicitly the physical limits of the system. One of the underlying assumptions is that the robot task can be smoothly tailored in space and scaled in time so as to fit to the system limitations. However, simply scaling of the task commands could recover motion feasibility but may no longer satisfy the tracking performance [9]. The above description requires off-line planning of the desired trajectory. This may result to be impractical in some applications, for example, when the desired trajectory is available while the robot is in motion or under unpredictable situations in human-robot interaction systems, in which the speed and acceleration magnitude become high during a short time interval. Online trajectory scaling schemes have been proposed in [10–13], mainly to satisfy torque limits. Moreover, simple hardware and/or software saturation of the commands results in the lack of execution of the desired motion [14].

Control of robots with bounded velocities and torques has proved to be challenging problems, and these have been addressed mainly for robot manipulators. For instance, on one hand, interesting contributions for robot manipulators with bounded torques are reported in [8–17]. On the other hand, control of robot manipulators with bounded velocities is considered in [18–25]. Control of mobile robots with bounded torques is reported in [26] and that with velocity constraints is reported in [27–31]. However, among all these results, a common characteristic is that the considered system is subject to different operation conditions, and its performance is deteriorated when the signals are saturated. Also, some of them do not completely guarantee the given limits during the system transitory stage, even when the initial conditions of the states are within the specified bounds. A formal stability analysis is missed in many of these works, and with few exceptions, the results are mainly valid for set-point regulation or for the case in which the kinematic model of the system is considered.

In this paper, we focus our attention on the motion control problem of the OMR subject to velocity constraints. Different from that reported in [27–31], the solution proposed here consists in the redesign of a well-known tracking controller to achieve global asymptotic tracking with desired bounded velocities, considering the dynamic model of the system. The proposed control scheme includes explicitly the desired bounds for the developed Cartesian velocities, which in turn bound the maximal velocities developed by the wheels. These bounds are freely set by the user considering the physical limits of the device. The proposal includes a tracking controller (TC) combined with a passive nonlinear controller (PNC) through a negative feedback connection structure. The TC is used to achieve the desired tracking performance, while the PNC is used to ensure bounded velocities. By using Lyapunov theory and passivity tools, global asymptotic tracking with desired bounded velocities is proved. Simulations and experimental results are provided to show the effectiveness of the proposed control strategy.

The rest of the paper is organized as follows. In Section 2, the dynamics of the considered mobile robot is given, and the problem formulation is stated. The main result of the proposed control system, and its stability analysis, is presented in Section 3. Simulations and real-time experiments are given in Section 4. Finally, some conclusions and future work ideas are given in Section 5.

#### 2. Problem Formulation

##### 2.1. Dynamic Model

Consider the OMR shown in Figure 1, whose dynamics is described by where is the configuration vector, which includes linear and angular positions of the center of mass of the OMR with respect to the inertial frame , is the velocity vector, is the acceleration vector, and is the control input which includes the torques applied to the wheels. is the inertia matrix, is the Coriolis and centrifugal force vector, and is the input matrix. These are given as