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Discrete-Time Modeling and Path-Tracking for a Wheeled Mobile Robot 1
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Modelado en Tiempo Discreto y Seguimiento de Trayectorias para un Robot Móvil Populsado por Ruedas
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Martin Velasco Villa* 2 , Eduardo Aranda Bricaire** and Rodolfo Orosco Guerrero***
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ESIME-Culhuacan, Instituto Politécnico Nacional Av. Santa Ana 1000, 04430 México D.F., México. velasco@cinvestav.mx **CINVESTAV-IPN, Departamento de Ingeniería Eléctrica, Sección de Mecatrónica Apdo. 14-740, 07000, México DF, México. earanda@cinvestav.mx *** Instituto Tecnológico de Celaya, Departamento de Ingeniería Electrónica, Av. Tecnológico y A. García S/N, Celaya Guanajuato, México. rodolfoo@itc.mx

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Article received on March 10, 2008; accepted on September 04, 2008
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Abstract The exact discrete-timemodel of a two wheel differentially driven mobile robot is obtained by direct integration of its continuous-time kinematic model. The discrete-time model obtained is used to design two discrete-time linearizing control laws with different singular manifolds. These control laws are used to propose a commutation control scheme that solves the path-tracking problem and guarantees internal stabilityof the closed loop system. The performance of the control scheme is evaluated by the real time implementation of the proposed control strategy over a laboratory prototype. Keywords: Discrete-time systems, mobile robots, feedback linearization, real time systems.
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Resumen El modelo exacto en tiempo discreto de un robot móvil propulsado por diferencias de velocidades es obtenido porintegración directa de su modelo cinemático en tiempo continuo. El modelo en tiempo discreto es utilizado para diseñar dos leyes de control por linealización por retroalimentación con diferentes variedades singulares. Estas leyes de control son usadas para proponer un esquema de control por conmutación que resuelve el problema de seguimiento de trayectoria y garantiza la estabilidad interna del sistema enlazo cerrado. El desempeño del esquema de control es evaluado mediante la implementación en tiempo real de la estrategia de control propuesta sobre un prototipo de laboratorio. Palabras clave: Sistemas en tiempo discreto, robots móviles, linealización por retroalimentación, sistemas en tiempo real.
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1 Introduction
Discretization of continuous-time systems is especially important forcontrol purposes because in actual practice most controllers are digitally implemented and therefore operate in discrete time. As a matter of fact, if an exact discrete-time model is available and a discrete-time control law can be synthesized from this model, then the performance of the controller is independent of the sampling rate. Unfortunately, the exact discretization of a continuous timenonlinear system is not always possible. A nonlinear system can be exactly discretized only in the case when the continuous-time model subject to piecewise constant input (sampled input) can be analytically solved. It is shown below that the kinematic model of a wheeled mobile robot satisfies this remarkable property. Feedback linearization is one of the control alternatives for nonlinear systems. Thereis a complete theory about this nonlinear control strategy [Isidori, 95; Nijmeijer and Van der Schaft, 90] dealing mainly with the continuoustime case. Several problems studied for the continuous-time case can also be stated on a discrete-time context [Kotta,
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First author work was partially supported by CONACyT-México, Under Grant 61713. M. Velasco-Villa is on sabbatical leave fromCINVESTAV-IPN, México.

Computación y Sistemas Vol. 13 No. 2, 2009, pp 142-160 ISSN 1405-5546

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95]. However, the solutions obtained for continuous-time systems are not always parallel to the solutions obtained for discrete-time systems. For the study of nonlinear discrete-time systems, see Nijmeijer and Van der Schaft,...