Proceedings of the XI DINAME, 28th February-4th March, 2005 - Ouro Preto - MG - Brazil Edited by D.A. Rade and V. Steffen Jr. © 2005 - ABCM. All rights reserved. CHAOS AND BIFURCATION ANALYSIS IN A COUPLED OSCILLATORS SYSTEM APPLIED IN THE LOCOMOTION OF A BIPEDAL ROBOT Armando Carlos de Pina Filho Universidade Federal do Rio de Janeiro, COPPE/PEM, Mechanical Engineering Department C.P. 68503, CEP. 21945-970, Rio de Janeiro, RJ, Brazil e-mail: pina-filho@bol.com.br Max Suell Dutra Universidade Federal do Rio de Janeiro, COPPE/PEM, Mechanical Engineering Department C.P. 68503, CEP. 21945-970, Rio de Janeiro, RJ, Brazil e-mail: max@serv.com.ufrj.br Luciano Santos Constantin Raptopoulos Universidade Federal do Rio de Janeiro, COPPE/PEM, Mechanical Engineering Department C.P. 68503, CEP. 21945-970, Rio de Janeiro, RJ, Brazil e-mail: Raptopoulos@aol.com Abstract. In this work, the central pattern generator (CPG), responsible for the production of rhythmic movements, is formed of a set of mutually coupled nonlinear oscillators of van der Pol. From a model of two-dimensional robot, oscillators with integer ratio of frequency were used for simulating the behavior of the hip angle and of the knees angles. Each oscillator has its own parameters and the link to the other oscillators is made through coupling terms. The objective of this work is to analyze the dynamics of this coupled oscillators system by using bifurcation diagrams and Poincaré maps. By means of the analysis and graphs generated in MATLAB ® , it was possible to evaluate some characteristics of the system, such as: sensitivity to the initial conditions, presence of strange attractors and other phenomena of the chaos, such as “crisis”. Based on the results of the study, we conclude that although the use of coupled oscillators represents an excellent way for generating pattern signals of locomotion, its application in the control of a bipedal robot will only be possible with the correct choice of parameters, which must be done from the data provided by the analysis of bifurcation and chaos. Keywords: bipedal locomotion, central pattern generator, chaos, nonlinear dynamics, oscillators. 1. Introduction The study of mechanisms that perform motor functions, in special, the study of mechanical members, intends not only construct autonomous robots, but also to help in the rehabilitation of people who have suffered some accident. The study of the locomotion is inserted in this context, and this has been intensively studied since the second half of century XX. An ample vision of the state of the technique up to 1990 can be found in works as Raibert (1986) and Vukobratovic et al. (1990). In the course of many years the human being has been trying, in all forms, to recreate the complex mechanisms that form the human body. Such task is extremely complicated and the results are frequently unsatisfactory. However, with the greater technological advances each time, based on theoretical and experimental researches, the man gets, in a way, to copy or to imitate some systems of the human body. It is the case, for example, of the central pattern generator (CPG), responsible for the production of rhythmic movements, such as to swim, to walk, and to jump, that it can be modeled by means of mutually coupled nonlinear oscillators. There are some significant works about the locomotion of vertebrates controlled by central pattern generators. Amongst them, Grillner (1985), Collins and Stewart (1993), and Pearson (1993) are very important. The human locomotion is partially controlled by a CPG, what can be evidenced in works such as Calancie et al. (1994) and Dimitrijevic et al. (1998). A correctly projected CPG can generate trajectories of reference for locomotion and can be used in the control of bipedal robots. In this work the CPG is formed of a set of mutually coupled nonlinear oscillators, in which each oscillator generates angular signals of reference for the movement of the legs. Each oscillator has its proper amplitude, frequency and parameters, and the linking to the other oscillators is made through the choice of coupling terms. We intend to evaluate the use of van der Pol oscillators. Some previous works about CPGs formed by van der Pol oscillators, applied in the locomotion of bipedal robots, can be seen in Bay and Hemami (1987), Dutra (1995), Zielinska (1996), Dutra et al. (2003) and Pina Filho (2004). The objective of the this work is to analyze the dynamics of this coupled oscillators system by using bifurcation diagrams and Poincaré maps. By means of the analysis and graphs generated in MATLAB ® , it was possible to evaluate some characteristics of the system, such as: sensitivity to the initial conditions, presence of strange attractors and other phenomena of the chaos, such as “crisis”.