Research Article Real-Time Inverse Optimal Neural Control for Image Based Visual Servoing with Nonholonomic Mobile Robots Carlos López-Franco, Michel López-Franco, Alma Y. Alanis, Javier Gómez-Avila, and Nancy Arana-Daniel Computer Science Department, CUCEI, University of Guadalajara, 44430 Guadalajara, JAL, Mexico Correspondence should be addressed to Carlos L´ opez-Franco; clzfranco@gmail.com Received 1 November 2014; Revised 21 January 2015; Accepted 21 January 2015 Academic Editor: Luis Rodolfo Garcia Carrillo Copyright © 2015 Carlos L´ opez-Franco et al. Tis 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. We present an inverse optimal neural controller for a nonholonomic mobile robot with parameter uncertainties and unknown external disturbances. Te neural controller is based on a discrete-time recurrent high order neural network (RHONN) trained with an extended Kalman flter. Te reference velocities for the neural controller are obtained with a visual sensor. Te efectiveness of the proposed approach is tested by simulations and real-time experiments. 1. Introduction Traditionally, robot motion control approaches have feedback provided by a taco-meter or encoder, whose advantages are its easy implementation and its low cost. However, in mobile robotics such information is not accurate due to the appear- ance of slip phenomenon. One sensor that can be used to over- come these problems is the visual sensor. In this work, we use computer vision techniques to overcome such disadvantages. Although a visual sensor is more accurate, it can suf- fer from unknown external disturbances due to the robot motion. In addition, the robot model is inaccurate and it suf- fers from parameter uncertainties. To overcome such prob- lems, we propose the use of a discrete-time recurrent high order neural network (RHONN) trained with an extended Kalman flter with visual feedback. Te main goal of optimal control theory is to determine the control signals that will force a process to satisfy physical constraints and minimize a performance criterion simul- taneously [1]. In optimal control theory, a cost functional is defned as function of the state and the control vari- ables. Unfortunately it requires solving the Hamilton-Jacobi- Bellman (HTB) equation, which is not an easy task. To avoid the solution of a HTB equation an inverse optimal control can be used [2]. In inverse optimal control, we start with the def- nition of a stabilizing feedback control, and then we have to show that it optimizes a cost functional. In this work, the input of the inverse optimal control is determined by visual feedback. Te visual sensor is responsi- ble of tracking the target and the estimation of the robot velo- cities to achieve the desired task. In our case the task consists in moving the robot from an initial pose to a desired pose with respect to a target object. 1.1. State of the Art. An extensive class of controllers have been proposed for mobile robots [3–9]. Most of these refer- ences present only simulation results and the controllers are implemented in continuous time. A common problem when applying standard control theory is that the required parame- ters are ofen either unknown at time or are subject to change during operation. For example, the inertia of a robot as seen at the drive motor has many components, which might include the rotational inertia of the motor rotor, the inertia of gears and shafs, rotational inertia of its tires, the robot’s empty weight, and its payload. Worse yet, there are elements between these components such as bearings, shafs, and belts which may have spring constants and friction loads [10]. 1.2. Main Contribution. Te paper main contributions are as follows: (1) presenting a controller for mobile robots which includes the robot dynamics and does not need the previous knowledge of robot parameters or model; (2) computing the trajectory references for the controller on real-time using Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2015, Article ID 347410, 12 pages http://dx.doi.org/10.1155/2015/347410