Handbook of Research on Advanced Intelligent Control Engineering and Automation

Transcription

1 Handbook of Research on Advanced Intelligent Control Engineering and Automation Ahmad Taher Azar Benha University, Egypt Sundarapandian Vaidyanathan Vel Tech University, India A volume in the Advances in Computational Intelligence and Robotics (ACIR) Book Series

2 Managing Director: Managing Editor: Director of Intellectual Property & Contracts: Acquisitions Editor: Production Editor: Typesetter: Cover Design: Lindsay Johnston Austin DeMarco Jan Travers Kayla Wolfe Christina Henning Michael Brehm Jason Mull Published in the United States of America by Engineering Science Reference (an imprint of IGI Global) 701 E. Chocolate Avenue Hershey PA, USA Tel: Fax: Web site: Copyright 2015 by IGI Global. All rights reserved. No part of this publication may be reproduced, stored or distributed in any form or by any means, electronic or mechanical, including photocopying, without written permission from the publisher. Product or company names used in this set are for identification purposes only. Inclusion of the names of the products or companies does not indicate a claim of ownership by IGI Global of the trademark or registered trademark. Library of Congress Cataloging-in-Publication Data Handbook of research on advanced intelligent control engineering and automation / Ahmad Taher Azar and Sunddarapandian Vaidyanathan, editors. pages cm ISBN (hardcover) -- ISBN (ebook) -- ISBN (print & perpetual access) 1. Automatic control. I. Azar, Ahmad Taher. II. Vaidyanathan, Sunddarapandian, TJ213.H dc This book is published in the IGI Global book series Advances in Computational Intelligence and Robotics (ACIR) (ISSN: ; eissn: X) British Cataloguing in Publication Data A Cataloguing in Publication record for this book is available from the British Library. All work contributed to this book is new, previously-unpublished material. The views expressed in this book are those of the authors, but not necessarily of the publisher. For electronic access to this publication, please contact: eresources@igi-global.com.

3 279 Chapter 10 Further Investigation of the Period-Three Route to Chaos in the Passive Compass-Gait Biped Model Hassène Gritli Institut Supérieur des Etudes Technologiques de Kélibia, Tunisia Nahla Khraief Ecole Supérieure de Technologie et d Informatique, Tunisia Safya Belghith Ecole Nationale d Ingénieurs de Tunis, Tunisia ABSTRACT This chapter presents further investigations into the period-three route to chaos exhibited in the passive dynamic walking of the compass-gait biped robot as it goes down an inclined surface. This discovered kind of route in the passive bipedal locomotion was found to coexist with the conventional period-one passive hybrid limit cycle. The further analysis on the period-three route chaos is realized by means of the Lyapunov exponents and the fractal Lyapunov dimension. Numerical computation method of these two tools is presented. The first return Poincaré map of the chaotic attractor and its basin of attraction are presented. Furthermore, the further study of the period-three passive gait is realized. The analysis of the period-three hybrid limit cycle is given. The balance between the potential energy and the kinetic energy of the biped robot is illustrated. In addition, the basin of attraction of the period-three passive gait is also presented. 1. INTRODUCTION Robotics has evolved in recent years in various fields such as industrial robotics, medical robotics, domestic robotics, military robotics, etc. One of the most interesting applications in robotics is the analysis DOI: / ch010 Copyright 2015, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.

4 Further Investigation of the Period-Three Route to Chaos of human walking through various prototypes of biped robots. The studies have been found to solve some problems related to the stability of human walking and also related to the design of active and passive prostheses of lower limbs of the human being. However, despite its simplicity, human walking is considered quite complex from a dynamic system point-of-view and it is not very well understood. The dynamic walking of biped robots is modeled by an impulsive hybrid nonlinear dynamics. To obtain a synergy between human walking gaits study and biped robots, a simple two-link bipedal mechanism will be a good basic experimental as well as theoretical model. In the concept of bipedal robotics, passive dynamic walking has attracted the attention of many researchers and has been considered as the starting point for the control of biped robots. Passive dynamic models of biped walking have proven useful in understanding generalized principles that govern walking motions. The term passive dynamics arises from the ability of these models to walk without active control. Thus, passive dynamic walking is a mode of bipedal locomotion for which the biped robot requires no exogenous source of energy but it only uses gravity to walk on an inclined plane (McGeer, 1990; Goswami et al., 1996, 1998; Wisse et al., 2004; Gritli et al., 2012c). This method of walk solves the problem of energy consumption of bipedal robots and get a maximum energetic efficiency. In addition, the use of passive dynamic walking is expected also to obtain additional insights into the design principles of legged locomotion in nature. The best known biped robot using passive dynamic walking is the compass-gait biped robot. This biped is a two-link bipedal mechanism that was originally studied in 1996 by Goswami et al. (1996, 1998). These researchers have shown that this type of bipedal walking can generate chaos and period-doubling bifurcations (Wiggins, 2003) based on certain intrinsic geometric and physical parameters of the biped robot. Until nowadays, many researchers are working on the passive dynamic walking of the compass-gait biped robot and other simple passive biped robots in order to find other properties that can help in the understanding of human walking and also for the control of walking gaits of biped robots. The list of publications is very long. We cite for example (Kaygisiz et al., 2006; Safa et al., 2007; Norris et al., 2008; Kai & Shintani, 2011; Li & Yang, 2012; Li et al., 2013). One of the most important factors for successful and efficient walking is stability. In fact, the physical biped robots that have been designed around passive dynamic walkers are very sensitive to small perturbations. One of fundamental tools used to investigate stability of dynamic systems is the Lyapunov exponents. Since the fundamental paper of Oseledec (1968), the Lyapunov characteristic numbers or exponents are an important tool for the characterization of dynamical systems attractors of finite-dimensional nonlinear dynamic systems and their initial sensitivity to nearby initial conditions. The spectrum of Lyapunov exponents measures in fact the average divergence or convergence of nearby orbits along certain directions in state space. Sekhavat et al. (2004) employed the concept of Lyapunov exponents in order to analyze the stability of nonlinear dynamical systems and showed that the method is constructive and powerful. The sign of the largest Lyapunov exponent can infer the stability of systems and can rigorously prove the stability of the nonlinear system if numerical artefacts are under control (Kuo, 1995; Sekhavat et al., 2004). We showed recently that a cyclic-fold bifurcation is generated in the passive dynamic walking of the compass-gait biped giving rise to a cascade of period-doubling bifurcations and then a period-three route to chaos (Gritli et al., 2012c). We have analyzed this period-doubling route to chaos by means of the spectrum of Lyapunov exponents and the fractal dimension (Gritli et al., 2012a). In this chapter, we will revisit the passive dynamic walking of the compass-gait biped robot. We will give further results on the period-three passive limit cycle and the corresponding route to chaos. Our analysis will be based first on the computation of the Lyapunov exponents and the fractal dimension in order to quantify order 280

5 Further Investigation of the Period-Three Route to Chaos and chaos (Gritli et al., 2014). We will present different attractive structures of the chaotic attractor. The Poincaré first return map of the chaotic attractor and its basin of attraction are also presented. Furthermore, we will study the period-three passive limit cycle using the state space and the balance of the kinetic and potential energy. We will study also the basin of attraction of the period-three passive gait compared with that of the conventional period-one passive gait. This chapter is organized as follows. In Section 2, an iterative review of chaos in passive dynamic walking of biped robots is presented. In Section 3, the compass-gait biped robot and its impulsive hybrid non-linear dynamics are presented. The passive walking patterns of the biped are also illustrated in this Section. Section 4 describes the analysis of the period-three route to chaos by means of the Lyapunov exponents and the fractal dimension. This section provide the calculation procedure of the spectrum of Lyapunov exponents and the Lyapunov dimension. Investigation of the chaotic attractor and the periodthree passive gait are presented in Sections 5 and 6, respectively. Section 7 contains the concluding remarks of this work. Finally, in Section 8, some thoughts for the future are given. 2. BACKGROUND More than a decade ago, there has been much interest stimulated in dynamic walking of passive biped robots. Tad McGeer (1990) showed that a completely unactuated and therefore uncontrolled leggedmachine can perform a stable walk on a range of shallow slopes. He is the first who introduced the idea of passive dynamic walking in robotic vocabulary. After the first passive dynamic walker made by McGeer, many follow-on researchers have resorted to theoretic as well as experimental studies of passive dynamic walking model. They have developed some prototypes of passive biped walkers (Goswami et al., 1996, 1998; Garcia et al., 1998, 2000; Osuka & Kirihara, 2000; Wisse et al., 2004; Collins et al., 2005; Zhang et al., 2008; Asano & Luo, 2009). The famous passive walkers are the point-foot walker, introduced by Garcia et al. (1998), and the compass-gait biped model which is studied by Goswami et al. (1996, 1998). The last prototype of biped robot is more realistic and more sophisticated than the point-foot walker. The passive gait model of the compass-like biped robot is qualified as an impulsive hybrid dynamic system defined by algebro-differential equations. Nonlinear phenomena in passive dynamic walking are very complicated, and their investigation has proceeded with difficulty. The most interesting phenomenon that passive-dynamic walkers exhibit is the period-doubling bifurcation route to chaos (Goswami et al., 1996, 1998). Goswami et al. were the first to report the occurrence of period-doubling bifurcation for the compass-like biped robot on a gentle slope leading progressively to chaotic motion. They have numerically showed that this biped robot exhibits period-doubling bifurcations by changing the slope angle and even parameters of the compass robot such as leg-mass location and legs mass. After that, Garcia et al. (1998, 2000) showed that the simplest walking model, the point-foot walker, and the knee model exhibit also period-doubling bifurcations and they showed the existence of both stable and unstable period gaits. In addition, Howell & Baillieul (1998) discovered a stable period-three gait and subsequent period-doublings leading to chaos for the point-foot walker without and with torso via homotopy method. Recently, chaos in the passive walking model of a point-foot walker has been investigated in (Li & Yang, 2012; Li et al., 2013). Iqbal et al. (2014) provide an overview of previous literature on the chaotic behavior of passive dynamic biped robots and its control. 281

6 Further Investigation of the Period-Three Route to Chaos We showed recently that a cyclic-fold bifurcation is generated in the passive dynamic walking of the compass-gait biped robot giving rise to a cascade of period-doubling bifurcations and then a route to chaos (Gritli et al., 2012c). This local-type bifurcation creates a pair of a period-3 stable gait and a period-3 unstable gait. Furthermore, we showed that the passive gait of a biped robot with unequal leg length exhibits also the cyclic-fold bifurcation with a hysteresis phenomenon (Gritli et al., 2011b). In addition, we showed in (Gritli et al., 2012a) that this walking locomotion exhibits two additional routes to chaos namely the interior crisis and the intermittency. We showed also in (Gritli et al., 2011b, 2012c) that the compass-gait biped robot falls down because of the birth of a global bifurcation known as the boundary crisis. This type of global bifurcation is generated mainly by means of the period-3 unstable limit cycle. In (Gritli et al., 2011a), we used an energy-tracking controller in order to stabilize and to track the period-three passive gait of the compass biped robot. In (Gritli et al., 2013), we controlled chaos exhibited in the passive dynamic walking of the compass-gait biped in order to obtain a periodone stable walking cycle. For a smooth dynamical system, several methods for calculating Lyapunov exponents have been well established (Parker & Chua, 1989; Ramasubramanian & Sriram, 2000; Lu et al., 2005; Chen et al., 2006). Many numerical algorithms allow an easy estimation of the spectrum of Lyapunov exponents for many smooth continuous systems described by differential equation of motion and for discrete maps described by difference equations. However, there is a numerical problem according to an overflow trouble due to the exponentially diverging solutions of a chaotic system. In order to keep the calculations away from this trouble, the Gram-Schmidth reorthonormalisation procedure is applied to the fundamental solution matrix (Parker & Chua, 1989). Therefore, after some integration steps, the normalization factors of the fundamental solution matrix are summed up over a long integration interval to calculate the Lyapunov exponents. Study of chaos in robotics by means of Lyapunov exponents is not much investigated. A few of work have been found in literature. For practical robotic systems, it is in general almost impossible to determine Lyapunov exponents analytically and they very often have to be calculated numerically. In (Yang & Wu, 2006; Sun & Wua, 2012), authors investigated the balance control and analyzed the stability of a biped during disturbed standing while satisfying the constraints between the feet and the ground. Recently, we used the spectrum of Lyapunov exponents in order to investigate order and chaos in the passive dynamic walking of the compass-gait biped robot and in the semi-passive dynamic walking of the torso-driven biped robot as they descend some inclined surfaces (Gritli et al., 2012a). 3. HYBRID DYNAMICS AND PASSIVE WALKING PATTERNS OF THE COMPASS-GAIT BIPED ROBOT 3.1. The Compass-Gait Biped Robot Figure 1 provides a diagrammatic representation of the compass-gait biped robot where the significant parameters in the dynamics description are given in Table 1 (Goswami et al., 1998; Gritli et al., 2012c). Such biped robot is a subclass of rigid mechanical systems subject to natural unilateral constraints. The compass-gait biped model is composed of two completely identical legs: a stance leg and a swing leg. 282

7 Further Investigation of the Period-Three Route to Chaos Figure 1. The model of the compass-gait biped robot down a slope φ. Table 1. Specifications of the compass biped simulation model Symbol Description Value a Lower leg segment 0.5 m b Upper leg segment 5.0 kg m Mass of leg 5 kg m H Mass of hip 10 kg g Gravitational constant 9.8m/s 2 In this biped model, the two legs are modeled as rigid bars without knees and feet, and with frictionless hip. Each leg has a mass m concentrated at a distance b from the hip of mass m H. For an adequate initial condition and a corresponding slope angle φ, the compass robot carries out a passive walk without any external action. This typical biped robot is powered only by gravity. The passive walk of such typical robot is constrained in the sagittal plane and is made up primarily of two phases: a swing phase and a very instantaneous impact phase. In the former case, the compass-like biped can be modeled as a double pendulum of two legs. The latter case occurs when the swing leg touches the ground and the previous support leg leaves the ground. The impact of the swing leg with the ground is assumed to be slipless plastic. The compass walk configuration is determined by the support angle θ s and the nonsupport angle θ ns. In Figure 1, α is the half-interleg angle. 283

8 Further Investigation of the Period-Three Route to Chaos 3.2. Impulsive Hybrid Nonlinear Dynamics It is well-known so far that the hybrid model of the passive dynamic walking of the compass-gait biped robot consists of nonlinear differential equations for the swing stage and algebraic equations for the impact stage (Gritli et al., 2012c, 2014) Continuous Dynamics of the Swing Phase T Let θ = θ θ ns s be the vector of generalized coordinates of the compass-gait biped robot. The motion of the compass-gait biped robot is described by the following Lagrangian system: ( ) + ( ) + ( ) = J θ θ H θ, θ G θ 0 (1) This continuous dynamics is in fact subject to unilateral constraints: { 0 1 } Ω = θ R 2 : ( θ ) = ( cos( θ + φ ) cos( θ + φ )) > h l s ns (2) with l = a + b. Matrices in (1) are defined by: J ( θ) = ( ) ( ) 2 mb mlb cos θ θ s mlb ( θ θ m l m l a s ns ) + + H cos ns 2 2 2, H ( θ, θ ) = 0 mlb sin( θ θ s ns ) mlb sin( θ θ θ s ns ) 0 θ 2 ns 2 s and G ( θ) = ( ) mb g sin θns m l + m a + l H ( ( )) sin( θs ) Algebraic Equations of the Impact Phase Algebraic equations of the impact phase are given by: θ θ = R θ e = S θ + + e (3) 284

9 Further Investigation of the Period-Three Route to Chaos where subscribes + and denotes just after and just before the impact phase, respectively. Matrices in (Eq3) are defined by: R e = and S Q 1 = Q e p ( θ) m ( θ ), ( ( )) ( ( )) mb b l cos θ θ ml l b cos θ θ ma m l s ns s ns H with Q θ p ( ) = 2 mb mbl cos( θ θ s ns ) ( ) mab mab m l mal θ θ H s ns and Q θ m ( ) = cos( ). mab 0 The impact phase occurs when θ and θ belong to the following impact surface: Γ = θ, θ, h ( θ) = l ( cos( θ + φ) cos ( θ + φ) ) = 0, h ( θ) = 1 s ns Passive Walking Patterns of the Compass Biped dφ dt ( θ) = 1 1 Φ ( θ) θ < 0 θ (4) The bifurcation diagram of Figure 2a provides all passive gaits of the compass-gait biped robot while walking down a slope (Gritli et al., 2012b, 2012c, 2013). Here, the bifurcation parameter is the slope angle φ. In this bifurcation diagram, the blue attractor A 1 is the conventional behavior exhibited by the passive compass-gait biped robot. A cascade of period-doubling bifurcations leads to chaos. However, the pink attractor A 2 is the one recently found by us (Gritli et al., 2012c). This attractor reveals a scenario of a period-3 route to chaos. Figure 2b shows this route. The attractor A 2 is born at the cyclic-fold bifurcation (marked as CFB). This bifurcation occurs at φ = At this bifurcation, a period-three stable periodic orbit (p3-spo) enters in collision with a period-three unstable periodic orbit (p3-upo). Such p-3 UPO is responsible on the generation of a double boundary crisis (marked as BC) which is found to be the main cause of the falling down of the compass-gait biped robot (Gritli et al., 2011b, 2012c). For the attractor A 1, the period-doubling to chaos is born from a period-1 stable gait (p1-spo). Chaos is found to be terminated by the boundary crisis at φ = However, in Figure 2b, the period-three stable gait undergoes a period-doubling leading to chaos. Chaos is dead for φ = Some periodic windows (PW) are appeared. The window which is marked by p9-pw in Figure 2b reveals the existence of a scenario of period-doubling to chaos originated from a period-9 passive gait (Gritli et al., 2012c). In the next of this chapter, we will interest in the analysis of the scenario of the period-three route to chaos presented in Figure 2b. This analysis is realized first by means of the spectrum of Lyapunov exponents and the fractal Lyapunov dimension. Secondly, we will analyze the chaotic attractor and the period-three passive gait. 285

10 Further Investigation of the Period-Three Route to Chaos Figure 2. Bifurcation diagrams: step period as a function of slope angle φ. (b) is an enlargement of (a) 4. ANALYSIS OF ORDER/CHAOS VIA LYAPUNOV EXPONENTS AND FRACTAL DIMENSION In the present section, we will analyze the variation of Lyapunov exponents and the fractal (Lyapunov) dimension in order to quantify order and chaos of the passive dynamic walking of the compass-gait biped robot. The spectrum of Lyapunov exponents is one of the most remarkable characterizations of dynamical systems attractors of a finite-dimensional nonlinear dynamic system of dimension n and their initial sensitivity for initial conditions. The determination of Lyapunov exponents provides a qualitative and quantitative characterization of the dynamic behavior through the exponential convergence or divergence of neighboring trajectories evolving in the phase space. A system with ordinary autonomous differential equations of dimension n has in fact n Lyapunov exponents (Parker & Chua, 1989). Positive Lyapunov exponents measure the average exponential spreading of nearby trajectories, and negative exponents measure the exponential convergence of trajectories onto the attractor. The sum of the Lyapunov exponents is the rate of the average volume contraction. In order that the solution of the differential system is an attractor, it is necessary that the sum of Lyapunov exponents is certainly negative. If the attractor is an equilibrium point, then all Lyapunov exponents are negative. If the attractor is a periodic oscillation, then there is at least one zero Lyapunov exponent. If it is periodic, the largest Lyapunov exponent is zero and the other n 1 Lyapunov exponents are negative. In the case of a torus, the attractor has ( ) another zero Lyapunov exponent and the remaining Lyapunov exponents are negative. If the attractor is chaotic, then there are one or more positive Lyapunov exponents. If there are several (higher than 2) positive Lyapunov exponents, then the attractor is hyperchaotic (Parker & Chua, 1989). Besides the Lyapunov exponents, the most fundamental property of a chaotic attractor that was considered is its fractal dimension. In fact, the identification of chaos includes finding a strange attractor in the dynamics of the state space, which is characterized by its fractal structure. The dimension of an attractor is a measure of the number of active variables and complexity of necessary equations to model the dynamics of the system dynamics (Sprott, 2003). Typical features of the fractal structure are: 286

11 Further Investigation of the Period-Three Route to Chaos They have a fine structure and no characteristic length scale, They are too irregular to be described by ordinary geometry, both locally and globally, and They have a certain degree of self-similarity, which means that small pieces of the object resemble the whole in some respects. Fractal structures can be either deterministic where they are precisely self-similar, or random they are only statistically self-similar. The self-similarity means that the structure is scale invariant. Geometric objects with fractional dimensions are called fractals, and in dynamical systems, it has been found that chaotic attractors are fractal objects. The determination of the fractal dimension is therefore a method for characterizing a chaotic attractor (Banerjee & Verghese, 2001). The dimension is a kind of quantifier describing the attractor from the geometrical aspects. The dimension of an attractor gives us an estimate of the number of active degrees of freedom in the system. If the dimension of the attractor is not an integer, hence the attractor is a strange attractor. Many methods have been proposed to characterize the fractal dimension of strange attractors produced by chaotic flows: the correlation dimension (Grassberger & Procaccia, 1983), the Hausdorff-Besicovitch dimension (Bergé et al., 1984), and the Lyapunov dimension (also called the Kaplan-Yorke dimension) (Kaplan & Yorke, 1978). In (Gritli et al., 2012a), we calculated the spectrum of Lyapunov exponents and the fractal dimension for only the attractor A 1. In the present section, we will compute them for the attractor A 2 that reveals the period-three route to chaos. In addition, since the passive dynamics of the compass-gait biped robot is four-dimensional, then we have four Lyapunov exponents Calculation of the Lyapunov Exponents Let x = θ θ is the state variable. The passive gait model of the compass-gait biped robot defined by (1) and (3) can be rewritten in the following state representation: T T T ẋ = f ( x)if x Ω (5) x = g ( x )if x Γ (6) + with Ω and Γ are defined by (2) and (4), respectively. The solution x of the impulsive hybrid nonlinear model (5)-(6) can be expressed in terms of flow as: x( t) = ϕ ( t, x0 ) (7) ( ) = ( ) with initial condition x t ϕ t, x For a smooth dynamical system which is represented by only the dynamics (5), methods for calculating Lyapunov exponents based essentially on the integration of a variational equation have been well 287

12 Further Investigation of the Period-Three Route to Chaos established (Parker & Chua, 1989; Ramasubramanian & Sriram, 2000; Lu et al., 2005; Chen et al., 2006). The variational equation of (5) is given by: δẋ J x δx = ( ) (8) f where J f is the Jacobian matrix of the function f. The variational equation (8) can be expressed in terms of the fundamental solution matrix Φ as follows (Parker & Chua, 1989): Φ ( t, x ) = J ( x) Φ( t, x ) (9) 0 f 0 ( ) =. where Φ t, x I The fundamental solution matrix Φ is called very often the sensibility matrix in the initial state x 0 (Hiskens & Pai, 2000) and is given by: t, x0 Φ( t, x0 ) = ϕ( ) x 0 (10) For continuous systems, the fundamental solution matrix Φ is obtained by integrating the following system: x f x t x J x t x Φ, Φ, f = ( ) ( ) = ( ) ( ) 0 0 x t x for ϕ, Φ( t, x ) = I = ( ) (11) However, impulsive hybrid systems defined by (5)-(6) exhibit discontinuities in the Jacobian matrix J f at the impact instant with the impact surface Γ. The jumps in the fundamental solution matrix will be computed analytically according to (Muller, 1995; Hiskens & Pai, 2000) as follows: ( ) = ( ) ( ) Φ + τ, x Π x, x + Φ τ, x (12) 0 0 where: ( ( ) = ( ) ( ) ( ) ( )) ( ) + ( ) ( ) + f x G x f x H x + Π x, x G x (13) T H x f x T g x with G ( x) = ( ) x h x and H ( x) = 1 ( ). In (12), τ is the impact instant. x 288

13 Further Investigation of the Period-Three Route to Chaos Therefore, calculation of the fundamental solution matrix Φ requires mainly a precise knowledge of the impact time τ and of course of the jump condition (12). Accordingly, in order to calculate the fundamental solution matrix Φ for the impulsive hybrid nonlinear dynamic (5)-(6), we must solve the dynamic system (11) and the following jump system: + x g x + + x x x x Φ τ, Π, Φ τ, = ( ) ( ) = ( ) ( ) 0 0 if x Γ (14) Contrary to periodic systems, there is a numerical problem for chaotic systems due to the exponential divergence of the fundamental solution matrix Φ. In order to remedy to this divergence problem, the Gram-Schmidth Reorthonormalisation (GSR) procedure is applied to the matrix Φ at some given time. For the case of discontinuous systems, the GSR procedure will be applied at impact instants τ. Then, the GSR procedure will be applied on the matrix Φ +. Therefore, the Lyapunov exponents λ i will be calculated as follows: λ i 1 + lim ( Φ τ, x vi ) (15) + 0 τ = ( ) τ with v i R 4 1 and the i th element of v i is 1 and the remaining elements are equal to Calculation of the Fractal Lyapunov Dimension The Lyapunov dimension is named also the Kaplan-Yorke dimension (Kaplan & Yorke, 1978). The fractal Lyapunov dimension is calculated using the spectrum of Lyapunov exponents as follows (Gritli et al., 2012a). Let λ 1, λ 2, λ 3 and λ 4 be the 4 Lyapunov exponents where λ 1 is the largest one and λ 4 is the smallest one. Let j be an integer such that: S j j = λ i 0 and S j = + λ i < 0 (16) 1 i = 1 j + 1 i = 1 Accordingly, the fractal Lyapunov dimension d L is given by: d L = j S j λ j + 1 (17) 289

14 Further Investigation of the Period-Three Route to Chaos 4.3. Spectrum of Lyapunov Exponents and Lyapunov Dimension for the Compass-Gait Biped Robot Variation of the spectrum of the Lyapunov exponents and the fractal Lyapunov dimension with respect to the slope angle φ are given by Figure 3a and Figure 3b, respectively. These diagrams are depicted for slopes between 3.85 and Recall that the cyclic-fold bifurcation is born at φ = , and chaos of the attractor A 2 is terminated at φ = The phenomenon of the period-three route to chaos occurs between and Outside of this range, the passive walking pattern is of period-1 (marked as p1-spo). Obviously, in this range, the first (largest) Lyapunov exponent λ 1 is zero while the three remaining Lyapunov exponents are negative for slopes between and In this interval of slopes, the passive gait of the compass biped robot is periodic and it exhibits a cascade of period-doubling bifurcations when the slope angle increases. The period-doubling (PD) phenomenon is expressed in each diagram with a cascade of parabolic curves. Furthermore, the attractor dimension remains almost constant at an integer value 1 when the gait is periodic except at the period-doubling bifurcations where the Lyapunov dimension is equal to 2. This translates the fact that the periodic attractor is dimensionally a line and all trajectories expand in state-space in only one direction. However, for slopes higher than , the largest Lyapunov exponent, λ 1 ( λ 2 ), oscillates between positive (zero) and zero (negative) values corresponding to chaotic and periodic gaits. The two other Lyapunov exponents, λ 3 and λ 4, remain always negative. The periodic gaits bring in fact about the existence of periodicity windows in the chaotic regime. In addition, we emphasize that λ 1 reaches its highest value which is equal to 0.3 at φ = Moreover, when the passive walking pattern is chaotic, the Lyapunov dimension is found to be higher than 2 and increases to reach its maximum fractal value 2.15 at φ = For a system with a four-dimensional state space, this indicates a strong phase-space volume contraction. It indicates also that the attractor is dimensionally close to a Euclidean plane. Figure 3. Variation of the spectrum of Lyapunov exponents (a) and the fractal Lyapunov dimension (b) as the slope angle φ increases for the period-three route to chaos 290

15 Further Investigation of the Period-Three Route to Chaos 5. ANALYSIS OF THE CHAOTIC ATTRACTOR In this section, we will analyze the chaotic attractor for the slope angle φ = The Chaotic Attractor and its First Return Map Figure 4 shows different forms of this chaotic attractor in different phase-spaces. The passive dynamic walking is described by a complete disappearance of order and is an extreme case of an asymmetric gait and affirming thus that the attractor is chaotic. Obviously, in Figure 4a and Figure 4b, the form of the chaotic attractor is quite attractive. Figure 4a shows the chaotic attractor in 2D plotted with respect to the angular velocities of the two legs of the compass-gait biped robot. The chaotic attractor seems like a butterfly. In addition, Figure 4b manifests another structure of the chaotic attractor plotted in the three-dimensional state-space. The chaotic attractor has the shape of a heart. The largest Lyapunov exponent and the fractal dimension were computed to be about 0.3 and 2.15, respectively. Figure 5 shows the Poincaré first return map in 2D. Figure 5b is an enlargement of Figure 5a. The first return map is composed of an infinite number of points irregularly distributed into three arcs of curves. By making an enlargement of the right part (Figure 5b), we can note that the first return map consists of several closed lines separated by empty spaces. This confirms that the attractor is well chaotic and it has a fractal dimension Basin of Attraction of the Chaotic Attractor Localization of the basin of attraction of some limit set is very important. It shows the set of admissible initial conditions allowing the gait of the biped robot to converge to some domain of attraction like the period-3 passive stable gait or the chaotic attractor. Furthermore, coexistence of two different attractors Figure 4. Chaotic attractor for the slope angle φ = in different state-spaces 291

16 Further Investigation of the Period-Three Route to Chaos Figure 5. First return map for the slope angle φ = (b) is an enlargement of (a) for the same set of parameters suggests a good knowledge of the basin of attraction of the two attractors and also a good knowledge of the boundary of each basin of attraction. Since the dynamics of the passive compass-gait biped robot is four-dimensional, graphical visualization of the basin of attraction is very difficult. Then, we should fix two state variables and vary the two others to obtain a 2D picture. Figure 6 shows the basin of attraction of the chaotic gait and that of the period-one gait. The pink set is the basin of attraction of the chaotic attractor. However, the blue set reveals the basin of attraction of the period-one gait. Here, the basin of attraction is plotted for a fixed initial position of the two legs. We have chosen θ ns = 21 and θ = θ 2 φ. It is obvious that the basin of attraction of the chaotic s ns Figure 6. Basin of attraction of the chaotic attractor compared with that of the period-one limit cycle for the same slope φ =

17 Further Investigation of the Period-Three Route to Chaos attractor is smaller than that of the period-one passive gait. Thus, a small perturbation on the initial conditions of the compass biped robot can bring the behavior of the period-one passive walk to another completely different form of locomotion, i.e. the chaotic gait. 6. ANALYZE OF THE PERIOD-THREE PASSIVE GAIT In this subsection, we focus on the analysis of the period-three passive gait of the compass biped robot. Then, we choose the slope angle φ = Period-Three Passive Gait in the Phase-Plane Figure 7 shows the angular velocity of each leg as a function of its angular position. In this state-space, small dark circles represent impact points of each leg with the ground. Whereas, small dark squares reveal states just after impact. Arrows are signs of behavior transition of each leg. The cycle of the passive gait consists of three steps in order that the compass robot returns to its initial state that is marked by solid squares. From an initial condition lying on the first cycle (marked by 1), the compass biped follows the cycles from 1 to 3 before it returns again to the first cycle. Table 2 summarizes some characteristics of the period-three gait. It is clear that, from the first step (cycle) to the third step, the three quantities (namely the step length, the average velocity and the mechanical energy) of the biped robot increases. We showed in (Gritli et al., 2011a) that the increase in the average velocity and in the step length could have a fundamental rule for the control of the bipedal walking of our compass biped robot to the period-three passive gait. Tracking of such period-three passive gait has in fact an important relevance. It can be employed as a reference trajectory that must be tracked in order to jump for example some wall, some obstacle, or some undesired territory. Furthermore, this period-three passive gait can be used also in order to speed up or to slow down the biped robot. Accordingly, for the control principle design of passive dynamic walker, Figure 7. A typical period-three passive limit cycle of the compass-gait biped for φ =

18 Further Investigation of the Period-Three Route to Chaos Table 2. Characteristics of the period-three stable passive gait Cycle Number Step Length (m) Average Velocity (m/s) Mechanical Energy (J) the aim is to make the walking pattern of a biped robot converges to a stable limit cycle or changes the original limit cycle to a new one by control input in order to take the biped robot to another walking plane. In (Gritli et al., 2011a), we realized the tracking control of the period-three passive stable gait by means of an energy tracking control law Energy Balance To investigate the period-3 gait, we refer to the energy balance in the passive gait of the compass biped robot. Figure 8 reveals the kinetic energy of the biped versus its potential energy during one cycle (which consists of three steps). As reported by Goswami et al. (1996, 1998), this energy balance plot consists of constant potential energies (dashed lines) translating the ground impact, and parallel inclined lines representing the swing phase. Moreover, each energy trajectory (line BC, or EF, or HI) is a straight line making a 135 angle with the kinetic energy axis. This affirms that the mechanical energy, which is the sum of the two energies: potential and kinetic, is constant during each step. The trajectory of the robot starts from the point A and follows the path ABCDEFGHI. The points C, F and I are the touchdown points of the swing leg with the ground. Horizontal lines CD, FG and IA represent the instantaneous loss of kinetic energy due to impact event with the ground. Figure 8. Potential energy as a function of kinetic energy of the compass-gait biped robot for the periodthree passive gait 294

19 Further Investigation of the Period-Three Route to Chaos Furthermore, it is shown in (Goswami et al., 1996, 1998) that there exists a limit cycle when the loss of kinetic energy at impact equals the change in potential energy during only one step. However, for the period-3 passive gait, this occurrence is not true. Indeed, the distance between an initial point (A or D or G) of a gait step and its end-point (D or G or A, respectively) is not equal to the distance CD or FG or IA, respectively. Nevertheless, the sum of lost kinetic energy is equal to the lost potential energy during a complete gait cycle that consists of three successive steps. For the studied period-three gait, the loss of kinetic energy during a gait cycle is calculated to be about J. The mechanical energy of the compass robot for the three steps is indicated in Table 2. Obviously, the mechanical energies increases from the first step to the third one Basin of Attraction of the Period-Three Passive Gait The basin of attraction of the period-three gait and that of the period-one gait are depicted in Figure 9. The blue set is the basin of attraction of the period-one gait, whereas the pink set reveals the basin of attraction of the period-three gait. The remaining white set refers to the initial conditions from which the compass robot falls down. The basin of attraction of the period-one gait is depicted for the same slope angle φ = Each basin of attraction is plotted for a two state variables of the compass-gait model. It is depicted for a fixed initial position of the two legs of the compass biped robot. We have chosen θ ns = 17 and θ = θ 2 φ, and we have varied the angular velocities of the two legs. s ns Obviously, in Figure 9, the basin of attraction of the period-three passive gait is very small compared to that of the period-one gait. Furthermore, a small change either on initial push or on the initial position of the biped robot can provoke a different behavior: the robot can fall down or can follow either the period-one gait or the period-three one. Figure 9. Basin of attraction of the period-one passive gait (blue) and the period-three passive gait (pink) 295

20 Further Investigation of the Period-Three Route to Chaos 7. DISCUSSION The focus of the work is a relative further study of chaos in robotics. We investigated the passive dynamic walking of the compass-gait biped robot on inclined slopes. The study was mainly articulated around the period-three route to chaos generated in the passive bipedal locomotion. Our further analysis on chaos and order was made by means of the spectrum of Lyapunov exponents and the fractal Lyapunov dimension. Computing method of these two tools was presented in details. We have demonstrated that the chaotic attractor can reach a fractal value about 2.15 and for which the largest Lyapunov exponent is calculate to be about 0.3. At these highest values, the bipedal chaos was found to be abruptly destructed by means of a boundary crisis. We showed that the fractal dimension reveals a strong volume contraction in the state space and the chaotic attractor is dimensionally close to a Euclidean plane. In addition, we have analyzed the first return Poincaré map of the chaotic attractor and its basin of attraction. Two different attractive beautiful forms of the chaotic attractor in the phase plane are illustrated. Moreover, we have analyzed some attractive characteristics of the period-three passive gait. This investigation was made with the state space, the balance of kinetic and potential energies, and the basin of attraction. We showed that the period-three passive gait can be employed as a target that should be tracked in order to increase the speed of the biped robot in only three steps and then in order to jump some undesirable obstacles. However, it is shown that the basin of attraction of this kind of passive gait is very small and any perturbation on the states of the biped robot can switch its actual bipedal locomotion to the conventional passive period-1 gait where its basin of attraction is too large. Then, stabilization and robustness of such period-three passive gait will be needed. 8. CONCLUSION In this chapter, our analysis on chaos was realized by means of the Lyapunov exponents and the fractal dimension. It is well known that these two tools are used to identify chaos and also to study stability of dynamic system. In our future direction of research, we will employ the spectrum of Lyaounov exponents in order to study stability of hybrid limit cycle of the passive gait of the compass biped robot and also other models of biped robots like the torso-driven biped robot. In addition, in our future works, we will interest in the computation of the stable and unstable manifolds of the chaotic attractor in order to demonstrate the existence of chaos in the passive dynamic walking of the compass-gait biped robot for some specified bifurcation parameter such as the slope angle. Furthermore, our intention is to use to center manifold theorem in order to transform the impulsive hybrid nonlinear dynamics of the compass-gait biped robot into a normal form. This will help us to study bifurcations and also chaos. REFERENCES Asano, F., & Luo, Z. W. (2009). On efficiency and optimality of asymmetric dynamic bipedal gait. In Proceedings of IEEE International Conference on Robotics and Automation, (pp ). Kobe, Japan: IEEE. doi: /robot

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