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1 INSTITUT NATIONAL POLYTECHNIQUE DE GRENOBLE N o attribué par la bibliothèque THESE EN COTUTELLE pour obtenir le grade de DOCTEUR DE L INPG ET DE L UNIVERSITE DE SANTIAGO DU CHILI Spécialité : Automatique-Productique préparée au Laboratoire d Automatique de Grenoble dans le cadre de l Ecole Doctorale- Electronique, Electrotechnique, Automatique, Télécommunications, et Signal et dans le Département de Génie Electrique présentée et soutenue publiquement par Eleuterio Claudio URREA OÑATE le 15 décembre 2003 Titre : CONTRIBUTION TO THE PROBLEM OF ORBITAL STABILIZATION: APPLICATION TO A FIVE DEGREES OF FREEDOM UNDERACTUATED ROBOT Directeurs de Thèse : M. C. CANUDAS-DE-WIT (LAG) Mme. I. MAHLA (USACH) JURY Mme. I. MAHLA Président M. M. DUARTE Rapporteur M. E. GONZALEZ Rapporteur M. C. CANUDAS-DE-WIT Directeur de thèse Mme. I. MAHLA Directrice de thèse M. H. KASCHEL Examinateur M. A. GUTIERREZ Examinateur

2 Abstract In this thesis we explore the orbital stabilization problem of a class of underactuated electromechanical systems. We address to systems in which the control design set up is different from typical formulations of output tracking and regulation, in which the set point is a priori given. We present a method and examples to achieve orbital stabilization on planar robots with a passive rotational first joint. The configuration of the whole electromechanical system is controlled with a reduced number of inputs. We also study the orbital stabilization in a bipedal robot. This is done considering gaits of a bipedal planar robot with one degree of underactuation. Our testbed has seven degrees of freedom, five rigid members connected to one another by purely rotational joint angles plus the Cartesian coordinates of the hips. In this study, the robot s gait includes the swing phase only 1, the stance leg acts as a pivot, and is no actuation at the end of this stance leg. Thereby, with this system we can investigate the problem of stable balancing in a bipedal robot. We provide a technique for choosing parametric outputs for the dynamic robot model to achieve stable balancing motions that exploit as much as possible the natural dynamics of the system. We present a general method for biped dynamic balance control. 1 i.e., only one of the lower limbs is on the ground surface at any given time. i

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4 Acknowledgements I would like to express my thanks to all the people who have helped me and supported me in my work. I would like to thanks especially my supervisors, Dr. Carlos Canudas who guided my work, and Dr. Ingeborg Mahla Alvarez who has always taught me to go until the tip of myself in her efforts to guide me. I am especially grateful for Xavier AGATI s boundless enthusiasm that never failed to motivate me in tough times (or in good times) for his encouragement, suggestions and unconditional support. I would like to thank Marcela Jamett, Francisco Sánchez, Pierre AGATI, Renée A., Marcel NOUGARET, Jacqueline N., John J. Martinez and his wife Lina E., Juan Carlos Avila, Salvador Carlos Hernández and the whole Laboratoire d Automatique de Grenoble tim, all provided excellent help and encouragement over the years. Finally, I would also like to thank the others members of my defense committee, Dr. Manuel Duarte, Dr. Héctor Kaschel, Dr. Alejandro Gutiérrez, and Dr. Eugenio González for their careful reading of my thesis on such short notice and for their constructive questions and comments. iii

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6 Objectives The primary goal of this thesis is to provide a systematic methodology for designing a feedback that transforms the full dimensional system motion into a one with lower dimension, where the limit cycles can be easily studied. The secondary objective is to develop a general technique for choosing parametric outputs of a system with one degree of underactuation to study its zero-dynamics, and to stabilize it. The tertiary goal is to study the orbital stabilization of a underactuated bipedal planar robot, to show stability and periodicity in this class of underactuated system. v

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8 Contents 1 Introduction Orbital Stability The Problem of Balancing Thesis Organization Divisions of the Bipedal Gait Cycle Locomotion Variables Background on Orbital Stabilization Conclusions Orbital Stabilization of Underactuated Systems: General Model Description General Model Control Objective Zero-dynamics Uniqueness of the Zero-dynamics Stable Periodic Solutions Orbit Generator Forced Zero-dynamics Reachability of O d Conclusions One-Link Model Study One-Link Robot Model: The Reaction Wheel Pendulum Target Orbits Zero-dynamics Adaptation Law Simulations Results Conclusions Two-Link Model Study Two-Link Robot Model: The Acrobat Robot Target Orbits Zero-dynamics Adaptation Law Simulations Results Conclusions vii

9 5 Tree-Link Model Study Three-Link Robot Model: The Gymnastic Robot Zero-dynamics Forced Zero-dynamics Simulations Results Conclusions Five-Link Model Study: The RABBIT Robot LAPIN: The Rabbit s Dynamic Model Simulator Basic Features Structure Transformation to Three-DOF Underactuated Mechanisms Simulations Results Conclusions Experimental Results on RABBIT Robot The Rabbit Robot Description Hardware Description Software Description Inner-Loop Comments on Experimental Results Conclusions General Conclusions and Future Research General Conclusions Conclusions Future Research A Résumé 81 A.1 Stabilisation Orbitale de Systèmes Sous-Actionnés A.1.1 La Stabilité Orbitale A.1.2 Le Problème du Balancement A.1.3 Modèle Général A.1.4 Objectif de Contrôle A.1.5 Dynamique Zéro A.1.6 Caractère Unique de la Dynamique Zéro A.1.7 Solutions Périodiques Stables A.1.8 Générateur d Orbite A.1.9 Dynamique Zéro Forcée A.1.10 Obtention de O d A.1.11 Conclusions A.2 Etude d un Modèle à Trois Articulations A.2.1 Modèle de Robot à Trois Articulations : Le Robot Gymnaste A.2.2 Dynamique Zéro A.2.3 Dynamique Zéro Forcée A.2.4 Résultats des Simulations A.2.5 Conclusions A.3 Commentaires sur les Résultats Expérimentaux sur le Robot Rabbit A.3.1 La Boucle Interne viii

10 A.3.2 Commentaires sur les Résultats Experimentaux A.3.3 Conclusions A.4 Conclusion Générale A.4.1 Conclusion A.4.2 Recherches Futures B Resumen 113 B.1 Estabilización Orbital de Sistemas Sub-Actuados B.1.1 Etabilidad Orbital B.1.2 El Problema de Balanceo B.1.3 Modelo General B.1.4 Objetivo del Control B.1.5 Dinámica Cero B.1.6 Unicidad de la Dinámica Cero B.1.7 Soluciones Periódicas Estables B.1.8 Generador de Orbitas B.1.9 Dinámica Cero Forzada B.1.10 Obtención de O d B.1.11 Conclusiones B.2 Estudio de un Modelo de Tres Eslabones B.2.1 Modelo del Robot de Tres Eslabones: El Robot Gimnasta B.2.2 Dinámica Cero B.2.3 Dinámica Cero Forzada B.2.4 Resultados de las Simulaciones B.2.5 Conclusiones B.3 Comentarios Sobre los Resultados Experimentales en el Robot Rabbit B.3.1 Lazo Interno B.3.2 Comentarios Sobre los Resultados Experimentales B.3.3 Conclusiones B.4 Conclusiones Generales B.4.1 Conclusiones B.4.2 Trabajo Futuro C Background on Stability 147 C.1 Stability Definitions C.1.1 Equilibrium point C.1.2 Stability C.1.3 Stability in the Sense of Lyapunov C.1.4 Uniform Stability C.1.5 Asymptotic Stability C.1.6 Uniform Asymptotic Stability C.1.7 Global Asymptotic Stability C.1.8 Exponential Stability, Rate of Convergence C.2 Lyapunov Stability Theory ix

11 D Equations of Motion for Three-DOF Model of Gymnastic Robot 153 D.1 Generic elements of the inertia matrix M(q) D.2 Generic elements of the Coriolis and centrifugal torques D.3 Generic elements of the equation which represents the zero-dynamics E Equations of Motion for Five-DOF Model of Rabbit 157 E.1 Kinematic Model E.2 Dynamic Model E.2.1 Impact Model E.3 Kinematic Equations E.3.1 Position Variables E.3.2 Velocity Variables E.4 Dynamic Equations E.4.1 Kinetic Energy E.4.2 Potential Energy Bibliography 168 x

12 List of Figures 1.1 Rabbit. The 5 DOF Biped Walking Robot Gait cycles of a bipedal walk Locomotion variables The first biped walking machine The highly successful Honda P-2 and P-3 bipedal robots Reaction Wheel Pendulum Reaction Wheel Pendulum system Series of consecutive positions showing the oscillations in the Reaction Wheel Pendulum for t = 2.25 to t = 25 sec Sequence of simulated movements showing the oscillations in the Reaction Wheel Pendulum for t = 0 to t = 2.25 sec Evolution of q p Evolution of q a Evolution of q a Phase portrait of the Reaction Wheel Pendulum. Convergence to an elliptic target orbit Evolution of b(t) and ḃ Acrobot. Super Mechano-Boy Acrobot system Sequence of images showing the oscillations in the Acrobot for t = 0 to t = 25 sec Evolution of q p (t) Evolution of q a (t) Evolution of q a Evolution of b Evolution of ḃ Convergence to an elliptic target orbit O d (x) Evolution of b(t) and ḃ Three-link planar robot Evolution of q p Evolution of q a1 (t) Evolution of q a2 (t) Evolution of q a Evolution of q a Phase portrait of the Gymnastic Robot. Convergence to an elliptic target orbit xi

13 5.8 Evolution of b 1 (t) Evolution of ḃ Evolution of b 1 (t) and ḃ Rabbit s model representation Block diagram for the LAPIN s structure Rabbit robot system: (a) 5DOF, (b) 4DOF, and (c) 3DOF Series of consecutive positions showing the oscillations in the Rabbit s model for t = 0 to t = 25 sec Evolution of q RH (t) Evolution of q RK (t) Evolution of q LH (t) Evolution of q LK (t) Evolution of q LH (t) Evolution of b 1 (t) Evolution of ḃ Convergence to an elliptic target orbit O d (x) Evolution of b(t) and ḃ Reference planes of the human body Rabbit s central column and circular path Rabbit s robot geometry FFT on angular position in joints: (a) Right hip, (b) Right knee, (c) Left hip, and (d) Left knee Rabbit Block Control The inverse pendulum equivalent system A.1 Rabbit. Le robot marcheur à 5-DDL A.2 Robot plan à 3 articulations A.3 Evolution de q p A.4 Evolution de q a1 (t) A.5 Evolution de q a2 (t) A.6 Evolution de q a A.7 Evolution de q a A.8 Portrait de Phase du Robot Gymnaste. Convergence sur une orbite elliptique désirée A.9 Evolution de b 1 (t) A.10 Evolution de ḃ A.11 Evolution de b 1 (t) y ḃ A.12 Plans de référence par rapport au corps humain A.13 Colonne centrale et barre radiale du Rabbit A.14 Géométrie de Rabbit A.15 Diagramme du système de contrôle de Rabbit A.16 Système de pendule inversé équivalent au robot B.1 Rabbit. El robot caminante de 5-GDL B.2 Robot plano de tres eslabones B.3 Evolución de q p B.4 Evolución de q a1 (t) xii

14 B.5 Evolución de q a2 (t) B.6 Evolución de q a B.7 Evolución de q a B.8 Plano de Fase del Robot Gimnasta. Convergencia a una órbita elíptica deseada B.9 Evolución de b 1 (t) B.10 Evolución de ḃ B.11 Evolución de b 1 (t) y ḃ B.12 Planos de referencia del cuerpo humano B.13 Puesto central y barra radial de Rabbit B.14 Geometría de Rabbit B.15 Diagrama en bloques del sistema de control para Rabbit B.16 Sistema equivalente de péndulo invertido E.1 Rabbit Model E.2 Rabbit Model. Torque convention xiii

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16 List of Tables 3.1 Parameters of the Reaction Wheel Pendulum Parameters of the elliptic orbit Initial conditions of the elliptic orbits Parameters of the Acrobot Parameters of the elliptic orbits Initial conditions of the elliptic orbits Parameters of the Gymnastic robot Parameters of the elliptic orbits Initial conditions of the elliptic orbits Parameters of the robot Parameters of the elliptic orbits Initial conditions of the elliptic orbits Main physical parameters on Rabbit Rabbit s inertia reducer parameters Association between the sign of the control torque/degrees and the action of motors Values of the dynamic parameters of PID controllers A.1 Paramètres du Robot Gymnaste A.2 Paramètres des orbites elliptiques A.3 Conditions initiales des orbites elliptiques A.4 Paramètres physiques de Rabbit A.5 Paramètres de l inertie des réducteurs de Rabbit A.6 Association entre les débattements articulaires de Rabbit B.1 Parámetros del robot Gimnasta B.2 Parámetros de las órbitas elípticas B.3 Condiciones iniciales de las órbitas elípticas B.4 Parámetros físicos de Rabbit B.5 Parámetros de inercia de los reductores de Rabbit B.6 Asociación entre el signo del control torque/grados y la acción de los motores.139 C.1 Summary of the basic theorem of Lyapunov xv

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18 Chapter 1 Introduction The term underactuation refers to the fact that not all joints or degrees of freedom of a system are equipped with actuators (i.e., less control inputs than generalized coordinates), or are directly controllable. Mechanisms that can perform complex tasks with a small number of actuators are desirable in view of their reduced cost and weight. Underactuated electromechanical systems is an interesting and growing area. It is receiving increasing attention in robotic applications, and it offers challenging control problems [GAP01, HC94]. Studying systems with passive degrees of freedom is of significance when: A robot is controlled in spite of actuator failure. Less power can be used for to operate it i.e., energy saving purpose. Some joints are intentionally designed to be passive for human safety consideration, for example, surgical robots. These nonholonomic electromechanical systems arise in several ways and broad applications, such as undersea and space robots, flexible robots, mobile robots, and other machines that imitate animal locomotion, as walking robots. Underactuated electromechanical systems generally have equilibria which depend on both their kinetic and dynamic parameters [DMO00]. Walking is one of the most common human actions, which requires complex control in order to achieve meaningful motion. Human bipedal walking is believed to be a largely passive activity which requires little active control [Mc90]. Legged locomotion has been used as a mechanism for transportation in biological systems for millions of years. Many years now, researchers have combined scientific observations of the agility and efficiency of legged animals with innovative engineering to construct many types of legged mechanisms. The study of electromechanical legged motion is mainly motivated by its benefits to locomotion in rough terrain, terrestrial exploration, mining, forestry, industrial automation, operation in hazardous environments and farming, as well as its potential benefits to prothesis development and testing [GAP01]. Biped mechanisms are naturally unstable systems. The control of these mech- 1

19 2 CHAPTER 1. INTRODUCTION anisms is a difficult and multidisciplinary problem which is regarded as the most crucial aspect of locomotion. Based on the dynamic features of the resulting locomotion, we can classify biped mechanisms as passive, static, dynamic or purely dynamic walkers. Passive walkers use Earth s gravity as a power supply because they can only walk downhill converting gravitational energy into kinetic energy. In a statically stable robot 1, the vertical projection of the center of gravity always remains within the convex region called support polygon 2. The locomotion strategy of static robots consists in planning robot motions that keep the projection of the baricenter always within the support foot while giving rise to negligible inertial forces. These robots can stop moving at any time in the gait 3 cycle and maintain balance. The main drawback of this kind of gait is the low locomotion speed and the necessity of large, active feet. Dynamically stable robots 4 utilize dynamic forces and feedback to maintain control, and they are stable in a limit cycle that repeats once each stride. This robot types achieve fast, natural motion via the principle of dynamic equilibrium, which was first applied to biped locomotion in [VJ69], with the introduction of the Zero Moment Point 5 (ZMP), subsequently used by other researchers, see [VBSS90, Go99]. With this method, the various limbs of the robot move so as to keep the ZMP, which accounts for inertial forces as well as gravitational forces, within the support foot. If the ZMP is in the support region, this means that the robot is always rotating around a point in the support region and the robot is considered dynamically stable. If this is not the case and the robot rotates around a point outside the support region, the supporting foot will tend to be removed from the ground or pressed into it, leading to instability. This criterion for stability cannot be applied on a robot that has no ankle joints. Purely dynamic walkers achieve locomotion without equilibrium neither static nor dynamic. This kind of bipedal robots have passive or no feet at all, and sometimes use an external support structure for lateral balancing. 1.1 Orbital Stability The study and understanding of oscillations are of great interest in many applications. For example, in bipedal robots many times periodic motions are desired. However, Lyapunov Stability Theory cannot be directly applied in stability analysis of walking robots because if, x(t) is a periodic solution of a pure autonomous robot (trajectory), and x(t + δ) another solution, for every value of δ, periodic solutions of an autonomous system cannot be asymptotically stable in the usual way [GEK97]. Therefore, is natural 1 Robots that only move through defined positions of stability in their walking gait and are never in an unstable configuration. 2 Region formed by the contact points of the feet on the ground. 3 Pattern of footsteps at a particular speed, or manner of walking or running. 4 Robots that move through unstable positions in their walking gait and need to intelligently adjust and plan their movements to remain stable at any given time. 5 Point on the ground around which the moment of gravity and inertial force become zero.

20 1.2. THE PROBLEM OF BALANCING 3 to define the stability of these system types in terms of its orbital stability [HM86, Ha85]. Definition. Stable Gait. A gait is stable if, starting from a steady closed phase trajectory C in a system (represented by Eq. (C.6)), any finite disturbance leads to another nearby trajectory of similar shape. Definition. Asymptotically Stable Gait. A gait is asymptotically stable if, it is stable, and in spite of the disturbance the system returns to the original cycle. Definition. Orbitally Stable Trajectory. The phase trajectory C in a system (represented by Eq. (C.6)) is orbitally stable if, given ɛ > 0, there is δ > 0 such that if, R is a representative point of another trajectory C which is within a distance δ of C at time t 0, then C remains within a distance ɛ of C for t 0. If no such δ exists, C is orbitally unstable. Definition. Trajectory Asymptotically Orbitally Stable. A trajectory C is asymptotically orbitally stable if, the trajectory C is orbitally stable, and the distance between C and C tends to zero as time goes to infinity. 1.2 The Problem of Balancing Biped robots form a subclass of legged or walking robots [GAP01]. Their main specificity is the intermittent contact with the ground, allowing them more versatility in theirs displacements, but resulting in a structural instability of these systems. Static balance in a gait requires the centre of mass to be over the supporting base at all times. The most walking and running gaits try to operate under periodic motions (i.e., balancing phases, walking, running, etc.). They require active balance in order to move because no matter which gait has been chosen, it will always involve the centre of mass to be outside the area of support. The problem of stable balancing can be considered as an interesting subtask to be performed a priori or a posteriori to a walking phase. This has motivated our work the orbital stabilization control problem of a seven-degree of freedom planar robot. In this study, we consider a prototype named Rabbit, see Fig This robot is modeled as a underactuated planar biped, which has seven degrees of freedom, five rigid links connected to one another by purely rotational joint angles plus the Cartesian coordinates of the hips. Rabbit has a trunk and two legs with knees but no feet. The axes between the trunk and each femur are actuated, as the axes of each knee, but the trunk is non actuated. The contact between the support leg and ground is modeled as a pivot, then the contact between leg and soil is prompt, and the model has five degrees of freedom. 1.3 Thesis Organization This thesis is divided into 8 chapters and organized as follows. First, we present a brief description about the orbital stability, the problem of balancing, the divisions of

21 4 CHAPTER 1. INTRODUCTION the bipedal gait cycle, the locomotion variables, and finally a brief background on orbital stabilization. Chapter 2 discusses the problem of orbital stabilization of underactuated mechanisms (with one degree of underactuation). It presents a general model, proposes a control law to reach a limit cycle (closed curve) that produces stable oscillating behaviour on the full hybrid model, and shows the existence of periodic orbits in underactuated systems. Chapter 3 presents an application of the control law proposed in Chapter 2 on the Reaction Wheel Pendulum. Chapter 4 contains an analysis of the application of the control method to produce stable oscillations associated to a limit cycle on the Acrobat Robot: Super Mechano-Boy. Chapter 5 describes an example of a 3-DOF Robot, the gymnastic robot, and simulation results are provided for parameters taken from a real-life model. In Chapter 6, previous results are extended on the dynamic model of a five-link prototype walker. Chapter 7 describes the application of the proposed control technique on a laboratory prototype known as Rabbit. Chapter 8 presents the main contributions of this thesis and discusses a number of possible directions for future work. Appendix B and Appendix A present spanish and french abstracts, respectively; Appendix C recall some background on control theory to understand this dissertation. Appendix D provides the dynamic equations of the presented example in Chapter 5. Finally, Appendix E concludes this thesis with the Rabbit s kinematic and dynamic equations. Figure 1.1: Rabbit. The 5 DOF Biped Walking Robot.

22 1.4. DIVISIONS OF THE BIPEDAL GAIT CYCLE Divisions of the Bipedal Gait Cycle The gait cycle is the period of time between any two identical events in the walking cycle. The gait cycle consists of two phases, stance phase (ST) and swing phase (SW), executed cyclically. ST is the period of time when the foot is in contact with the ground, constituting 62 % of the gait cycle. SW is the period of time when the foot is not in contact with the ground, constituting the remaining 38 % of the gait cycle. The period of time when both feet are in contact with the ground is called double support (DS). DS occurs twice in the gait cycle, at the beginning and end of the stance phase. Single support (SS) is the period of time when only one foot is in contact with the ground. As depicted in Fig. 1.2, in walking, this is equal to the swing phase of the other limb. Initial contact (IC) is the point in the gait cycle when the foot initially makes contact with the ground. IC represents the beginning of the stance phase. Heel contact (HC) is the IC made with the heel. T erminal contact (TC) is the point in the gait cycle when the foot leaves the ground. TC represents the end of the stance phase or beginning of the swing phase. T oe-off (TO) is the terminal contact is made with the toe. During the gait cycle, the mass center of the entire body moves up and down in a sinusoidal pathway of low amplitude. As reference, in a human, this is typically about 3.5 cm for an averaged height individual. Phases Periods Events Percent of cycle ST SW DS SS DS SS DS Right Left Left Right Right Left HC TO HC TO HC TO 0 % 50 % 100 % SW ST Figure 1.2: Gait cycles of a bipedal walk. 1.5 Locomotion Variables The motion of the biped robots is completely characterized in terms of the following variables: length of the step Ls, hip height Hh, maximum hip ripple Hr, maximum clearance of the swing limb Hm, and leg link lengths L and l. The hip height is defined as the mean height of the hip along the cycle of walking. The hip ripple is measured as the peak-to-peak oscillation magnitude, see Fig. 1.3.

23 6 CHAPTER 1. INTRODUCTION Ls Hr L Hh l Hm Figure 1.3: Locomotion variables. 1.6 Background on Orbital Stabilization Over the last few decades many researchers have presented diverse mathematical models, control techniques, designs and developments of bipedal robots. A bipedal robot may be viewed as a highly nonlinear multi-link system that can ambulate on a given walking surface and to interact with its environment through its feet. From practical experience as a human, to avoid falling down on bipedal robots, good balance is required. Achieved this task with only two legs presents complex control problems. Bipeds fall within a special class of systems naturally unstable, thus a thorough stability analysis needs the inclusion of nonlinear tools. Orbital stabilization problem has attracted a great deal of attention in the last years. In inverted pendulums and bipedal robots the oscillations are not always present in the open-loop system, then a feedback law leading a stable closed-loop system is required. Such systems that generate their own stable periodic motions are desired because in bipedal robots there is a need for making the walking and/or running gaits varying on-line [CEU02]. In this section we review some techniques for bipedal robot control and the main results in orbital stabilization which could be applied to control such systems. Fig. 1.4 depicts the first biped walking machine: The Steam Man by Georges Moore in 1893 [Ro94]. The delay in the development of biped robots, with regard to others legged robots (four and six legged), is due to problems of balance. The issue of balance is avoided with static walkers and crawlers, see Fig Many of the early biped walkers moved slowly and/or had large feet. Stability with these bipeds is achieved passively 6. 6 Static walking assumes that if at any time the robot s motion is stopped, it will stay indefinitely in

24 1.6. BACKGROUND ON ORBITAL STABILIZATION 7 Active balance allows for stability with smaller feet and for dynamic walking and running. Figure 1.4: The first biped walking machine. BIPER-3 is the first bipedal robot to be successfully created in 1984 to use true dynamic balance [MS84], which was modelled after a human walking on stilts. It had only three actuators, one to change the angle separating the legs in the direction of motion, and the remaining two which lifted the legs out to the side in the lateral plane. Figure 1.5: The highly successful Honda P-2 and P-3 bipedal robots. Kajita et al. [KYK92] used analytical techniques for designing and controlling robot motion. They controlled bipedal dynamic walking by restricting the movement of the center of mass (COM) in an ideal sense to the horizontal plane only. This motion was termed a potential energy conserving orbit and could be expressed by a simple linear differential equation, which simplified the calculations involved. This technique had the tendency to produce complex equations governing the motion of the robot, which often a stable position.

25 8 CHAPTER 1. INTRODUCTION had no solution and had to be approximated or linearized. Other similar analytical approaches actually increased the complexity of the problem by introducing new links to the biped model. Takanishi et al. [TLTK90] used the robot WL-12RIII with a control system which manipulated the zero moment point (ZMP) to achieve dynamic stability, even on uneven surfaces. This robot had seven links including a trunk or upper trunk link with two degrees of freedom, thus allowing it to pitch and roll relative to the forward direction of the robot. Hauser and Choo [HC94] presented a technique to construct an autonomous Lyapunov function for exponentially stable periodic orbits. They showed that a periodic orbit is exponentially stable if and only if, the linearization of the dynamics transverse to the orbit are exponentially stable. The methodology consist in constructing a special set of local coordinates (angle and radius) around a periodic orbit that essentially separates the tangential and transverse dynamics of the system, and then constructing an autonomous Lyapunov function that proves the exponential stability of the periodic orbit. This methodology does not propose any procedure to assure the existence of a limit cycle, however, the control design is based on the existence of a limit cycle. Kajita and Tani [KT91, KT95] introduced the Method of the Inverted Pendulum (IPM). The main advantage of this method is its simplicity and its closed analytic solution. The IPM assumes a concentrated mass at the trunk and neglects all other masses. Particularly, the legs are supposed to be of no mass, which is more or less a rough simplification. Pratt et al. [PDP97] used a technique that employs virtual springs and dampers to describe interactive force behaviour to control their two-dimensional walking robot, the Spring Flamingo. In employing this technique, called V irtual M odel Control, the robot behaves as if those components were in fact attached to it. The virtual components apply virtual forces, which are transformed into real torques at the joints of the robot via Jacobian transformation matrix. The advantage of this technique is that the controller is mostly intuitive and easy to understand. Advanced versions of the IPM have been proposed by Park and Kim in [PK98], and Park and Cho in [PC00]. Instead of only considering a concentrated trunk mass, they also consider a concentrated mass for the swinging leg. However, the movement of the swinging leg is prescribed and the trajectory of the trunk is adjusted only with respect to the gravitational and hence static effect of the swinging leg. If a controller is based upon tracking, the disturbances on the system cause delayedaction with respect to the planned motion because the feedback system is obliged to play catch up in order to synchronize again the motion with the reference trajectory. A solution for this problem is to impose the robot walking motion converges to a periodic orbit (cycle limit). To accomplish this, we can parameterize this orbit with respect to the state of the robot instead of time, then the output system is formed and zeroed by a feedback controller law employing the concept of virtual constraints. This problem has been studied in Canudas-de-Wit et al. [CEU02, UCM02], where an appropriate feedback is proposed in order to transform the full dimensional system into one with lower dimension (zero-dynamics). Then, a controller is designed such that the low-dimension subsystem reaches a specified target orbit.

26 1.6. BACKGROUND ON ORBITAL STABILIZATION 9 Afterwards, Grognard and Canudas-de-Wit [GC02] proposed a methodology for the design of limit cycles in systems having more inputs available than outputs to regulate. The primary outputs needed to be regulated as a requirement of the given control problem, and the secondary outputs were regulated in order to force the system to present limit cycles. In this way, the system was controlled by using input-output feedback linearization. This methodology was illustrated on the induction motor, and conditions were given to assure that a globally attractive limit cycle in the zero-dynamics results in a globally attractive limit cycle in the whole system. For achieve stable and robust oscillations around the upright position on the Furuta pendulum, Aracil et al. [AGA02] associated the system oscillations to a limit cycle by means of a supercritical Hopf bifurcation in a generalized hamiltonian system (target system). The control synthesis is accomplished in two parts, the first step is to partially linearize the open-loop system, and then to define a partial linearizing controller to match a desired closed-loop oscillatory structure. This method works well for fully actuated second-order systems, and for certain underactuated ones, however, this method does not assure the stability of all system state variables. In [AGG03] Aracil et al. used back-stepping for controller orbital stability on full actuated systems, and the stability and robustness of these systems are guaranteed by means of a Lyapunov function. Muñoz-Almaraz et al. [MFG03] applied a method to characterize the bifurcation diagram of the Furuta pendulum in absence of external forces when the energy of the system is varied. This pendulum was modeled as a two degrees of freedom integrable Hamiltonian system because the system is invariant under rotations around the axis in absence of external forces. Then, the bifurcation diagram is organized around solutions that are invariant under the symmetry and bridges connecting different subharmonic bifurcations. The result of the application of a continuous symmetry on a generic periodic solution is another periodic solution with identical stability properties. This method is applicable on unforced and autonomous systems because under control action or time dependence, in general, the Hamiltonian character of the system equations is lost. For obtain locally asymptotically stable limit cycles on second order systems with harmonic target orbits, Vivas and Rubio [VR03] present a setup for the computation of nonlinear state-feedback H controls. The controller synthesis was formulated to induce a certain L 2 -Gain attenuation level for a mapping between the disturbances acting on the system and an error signal that weights the control action and deviations from the harmonic target orbit. This work gives an indication of the performance of the system, but it only can be extended to a higher order underactuated systems that can be posed in strick-feedback form. Perram et al. [PSCG03] presented an explicit form of a general integral of motion for the dynamics of any n-degrees of freedom electromechanical system with n 1 holonomic constraints. These holonomic constraints can be considered as artif icial constraints. They presented a methodology to construct cycles in electromechanical systems, and to achieve locally orbital stabilize these cycles. The effect of different holonomic constraints on the zero-dynamics and the general integral were given for the cart-pendulum. An algorithm based on the back-stepping idea has been proposed for stabilize the cycles

27 10 CHAPTER 1. INTRODUCTION when the electromechanical system does not satisfy the holonomic constraint.

28 1.7. CONCLUSIONS Conclusions Humans present a very elegant model of locomotion to emulate. They have the ability to adjust their gait in the presence of pathological conditions in order to maintain their locomotive functionality. Some seemingly simple behaviours such as human walking are difficult to model because of their inherent instability. Legged robots offer several potential advantages over wheeled vehicles, and their use is very promising in the human environment. The study and understanding of oscillations are of great interest in many applications. Such oscillations can be found in bipedal robots when a periodic motion is desired. Bipedal robots differ from others legged robots because their structure is inherently unstable no matter how they are designed. The stability of their motions is a crucial topic, which was analyzed from the very beginning of research on bipedal robots and is still under way. Minimizing actuation and control on bipedal robots, we have an approach to the imitation of human motions. The design and construction of feasible walking and running machines remains a challenge. Thus in bipedal research, an understanding of active balance is required.

29 12 CHAPTER 1. INTRODUCTION

30 Chapter 2 Orbital Stabilization of Underactuated Systems: General Model Description In this chapter we present the research about the orbital stabilization of general electromechanical systems of n-dof with one degree of undeactuation. Many of the walking machines developed have large feet in order to allow substantial torque to be generated by motors located at the ankles. One approach to avoid this problem is to balance the system with only actuation at the knee and leave the ankle unactuated. This has motivated our study of the orbital stabilization problem of underactuated systems 1. In the underactuated robots, due to the presence of passive joints, their dynamics is considerably more complex in comparison with fully actuated systems. 2.1 General Model We consider a class of nonholonomic jointed underactuated electromechanical systems of n-dof where the generalized forces are only actuation torques/forces, without flexibilities and no other potential based actions occur. The dynamics of this system is given by the follow Lagrange equation: M(q) q + C( q, q) q + g(q) = B τ (2.1) where q R n is the joint position vector, M(q) R nxn is the inertia matrix, C(q, q) R n are the Coriolis and centrifugal torques, g(q) R n is the gravity vector, B is a constant matrix of rank m, and τ R m is the actuation vector which includes all external generalized forces, with m < n the number of actuators. B can be defined as: 1 Systems with passive degrees of freedom that exhibit a nonholonomic behaviour. 13

31 14 CHAPTER 2. ORBITAL STABILIZATION OF UNDERACTUATED SYSTEMS: GENERAL MODEL DESCRIPTION B = [ 0(n m, m) I m ] (2.2) Denote N(q, q) = C( q, q) q + g(q), and assume that the state-space can be transformed and partitioned via a diffeomorphism φ into an m-dimensional part and a (n m)- dimensional part; then the configuration space q can be partitioned in two sets of coordinates (q a, q p ), therefore (2.1) can be expressed as: M p (q) q + N p (q, q) = 0 (2.3) with: M a (q) q + N a (q, q) = τ (2.4) q = [ qp q a ] (2.5) M(q) = [ Mp (q) M a (q) ] (2.6) N(q, q) = [ Np (q, q) N a (q, q) ] = [ Cp (q, q) + g p (q) C a (q, q) + g a (q) ] (2.7) where q p R n m and q a R m are the angles of the passive and the active joints, respectively. In general, the subscript p is used to represent quantities relative to the passive joint, whereas a represents this for the active joints. Eq. (2.3) shows that the torque of the passive joints is zero, and represents a set of (n m)-second-order differential constraints on the system, i.e., impose restrictions on the admissible generalized accelerations for any actuation command. These constraints includes generalized coordinates q, velocities q, and accelerations q. Eq. (2.4) describes the dynamic in relation to the torque of the active joints, τ R m. 2.2 Control Objective The control objective is to bring the system (2.1) to its zero-dynamics, and to stabilize it, because stable periodic orbits of the zero-dynamic correspond to stabilizable orbits of the full model (2.1). We can specify a convenient set of outputs in a such a way that zeroing this outputs would be equivalent to achieve the desired robot configuration. For this system, we define the following output m-dimensional: y(q, t) := q a φ(q p, θ(t)) (2.8) where φ(q p, θ) is a smooth function, and θ is a parameter vector allowed to vary with the time, that will become the control used to set a particular zero-dynamics. Since

32 2.3. ZERO-DYNAMICS 15 dim(q a ) = dim(τ), all the available actuation forces/torques have to be used for that purpose. Defining J(q) = y, doing the following manipulations from (2.1): q q + M 1 N = M 1 B τ (2.9) J q + J M 1 N = J M 1 B τ (2.10) τ = (J M 1 B) 1 [J M 1 N + J q ] (2.11) and assuming that JM 1 B is not singular, at least locally, the following feedback law: [ τ = (J M 1 B) 1 u + J M 1 N J ] q γ (2.12) with 2 γ = J θ + J θ, linearizes the output y i.e., ÿ = u. Therefore, the partial decoupling/feedback linearization of (2.1), can be done using the control (2.12). We can take many output feedback structures to ensure that y 0 asymptotically or in finite-time 3, then a great family of non-smooth feedbacks functions can be also designed in order to drive the system to the zero-dynamics [Ka96]. In this study we chose the following family of non-smooth feedbacks functions: where 0 n < 1, k, and k s are positive constants. u = λ ẏ k s s k s n sign(s) (2.13) s = ẏ + λ y (2.14) ẏ = q p + J q + φ θ θ (2.15) 2.3 Zero-dynamics The zero-dynamics is defined as the internal dynamics of the system when the required initial conditions and controls are applied to keep the outputs zero for all time [HY99]. The zero-dynamics of (2.1), i.e., (2.3), can also be viewed as a non-holonomic constraint associated with the actuated part of the system. The dynamics (2.1) projected on the constraint y(q, t) = 0, that implies ẏ = ÿ = 0 and u = 0, is called the zero-dynamics associated with φ. After substituting (2.11) in (2.1), and doing the following manipulations: M q + N B (J M 1 B) 1 [J M 1 N + J q ] = 0 (2.16) M q + N B (J M 1 B) 1 J M 1 [M q + N] = 0 (2.17) [ In B (J M 1 B) 1 J M 1] [M q + N] = 0 (2.18) 2 γ is the other terms involved in the differentiation of (2.8), that introduced in (2.1) gives ÿ = u. 3 Convergence is suited to avoid problems due to finite time escape, and hence to ensure system internal stability.

33 16 CHAPTER 2. ORBITAL STABILIZATION OF UNDERACTUATED SYSTEMS: GENERAL MODEL DESCRIPTION we can obtain this zero-dynamics (Eq. (2.18)). Defining: P (q) := [ I n B (J M 1 B) 1 J M 1] (2.19) It can be rewritten (2.18) as: P (q) [M(q) q + N(q, q)] = 0 (2.20) where P (q) is the projection operator on the kernel of J M 1 in the orthogonal direction of B. The zero-dynamics (2.20) lies then on a configuration space of dimension n m Uniqueness of the Zero-dynamics From Eq. (2.1) we have: [ ] q q =: M 1 (q) + [ C(q, q) q g(q) + B τ] (2.21) Differentiating the output twice, from (2.21) we have: ÿ = L 2 f h(q, q) + L g L f h(q) u (2.22) The invertibility of the coupling matrix L g L f h(q) at a given point assures the existence and uniqueness of the zero dynamics in the neighborhood of that point [Is95]. 2.4 Stable Periodic Solutions Our objective is to find a smooth function φ and an adaptation law for θ by means of a dynamic feedback, such that the zero-dynamics exhibits stable periodic solutions. If we consider systems where the number of actuators is m = n 1, we will have a zerodynamics of dimension one (dim(q p ) = 1) described by a second order nonlinear equation. Then, its behaviour can thus be studied in the plane. If additionally we consider a 2-DOF system, and we take a linear constraint of the form q a = aq p + b(t), and the following constraints: q a = φ(q p, θ) (2.23) q a = J p (q p, θ) q p, +J θ (q p, θ) θ (2.24) q a = J p (q p, θ) q p + r p (q p, q p, θ, θ, θ) (2.25)

34 2.4. STABLE PERIODIC SOLUTIONS 17 where: θ = θ = [ a b [ 0ḃ ] ] (2.26) (2.27) θ = [ 0 b ] (2.28) J p = φ q p (2.29) J θ = φ θ (2.30) r p = J p q p + J θ θ + J θ θ (2.31) by taking (n m) independent lines of the following system, the dynamics (2.20) can be written as a function of the unactuated joints q p only. [ [ ] ] Jp q P (φ(q p ), q p, θ) M(φ(q p ), q p, θ) p + r p + N(φ(q q p ), q p, J p, q p, J θ, θ, θ) = 0 (2.32) p By defining: [ q T p ] z = (2.33) q T p we can present the zero-dynamics (2.32) in a convenient set of local coordinates, i.e., ż = f(z, φ(z, θ), θ, θ, θ) (2.34) In this state-space representation, the vector θ gives us an additional degree of freedom that is used to produce a stable oscillatory behaviour in (2.34). If the elements of θ have a constant or periodic dynamics, in Eq. (2.34) we can impose a periodic motion on z by means of the restriction imposed by y = 0. Then, the zero-dynamics acts as a nonlinear autonomous oscillator that drives the rest of the system coordinates to a periodic orbit, by means of the constraint imposed by the actuated joint coordinates q a = φ(q p, θ). In this case, the equation (2.32) takes the particular form: β 2 (q p, φ, θ) q p + β 0 (q p, q p, φ, θ, θ, θ) = 0 (2.35) q p O p Q P, where Q P is the workspace for the underactuated variable, and θ (resulting from the adaptation law θ) φ, the solution of (2.35) is well defined if: β 2 (q p, φ, θ) > 0 (2.36)

35 18 CHAPTER 2. ORBITAL STABILIZATION OF UNDERACTUATED SYSTEMS: GENERAL MODEL DESCRIPTION By defining q d p as the orbit center (desired equilibrium), x = [ x1 x 2 ] (2.37) with: x 1 = q p q d p (2.38) and x 2 = q p, then we can write (2.35) as: β 2 (x 1, q d p, φ, θ) ẍ 1 + β 0 (x, q d p, φ, θ, θ, θ) = 0 (2.39) that under assumption (2.36) has the following state-space representation x = f(x, θ, φ, θ, θ): ẋ 1 = x 2 (2.40) ẋ 2 = β(x, φ, θ, θ, θ) with β = β 0 β Orbit Generator The problem addressed now, is to introduce the generalized target orbit O d (x) defining a closed path in the phase plane. Let also O d (x) define an invariant and attractive set of the solution, at least locally, of the generalized orbit generator, ẋ 1 = x 2 ẋ 2 = β d (x) (2.41) Thereby, we assume that the function β d (x) vanishes at the equilibrium x = 0 (β d (0) = 0) only, which is included in a convex set where O d (x) is assumed to be. In other words, there exist a closed set M R 2, s.t. M contains no equilibrium points and is positive invariant. The bounded semi-positive orbit O d (x) is thus contained in M. 2.6 Forced Zero-dynamics Once reached the zero-dynamics, its motion governed by (2.20) is free since no more control is available. In many cases, this free motion is a periodic orbit [ABEDM01]. If we consider that the system (2.1) has one degree of underactuation (m = n 1), then its zerodynamics can be expressed using a single coordinate denoted by x 1. If the phase portrait of this system is a closed curve O (orbit), then this periodic orbit which characterizes the zero-dynamics can be uniquely specified by a trio (φ, x 1, x 2 ). The idea now is to specify

36 2.7. REACHABILITY OF O D 19 such a periodic orbit as a final goal, recalling that we can consider the choice of φ as a way to modify it. For doing that, we will consider the following centered family of elliptic target orbits 4 : O d = { x : V d = 1 } 2 (α d x x 2 2) (2.42) as a function of the parameters set {V d, α d, q d p}, where V d is the desired orbit level, α d is the desired orbit shape, and q d p is the desired orbit center. These orbits attract the solutions of the generalized orbit generator (2.41), with β d (x) defined as: β d (x) = α d x 1 + k V x 2 Ṽ (x) (2.43) where the energy level of the orbit, denoted by Ṽ (x), is given by: this is: Ṽ (x) = V (x) V d = 1 2 (α d x x 2 2) V d (2.44) ẋ 1 = x 2 ẋ 2 = α d x 1 k V x 2 [ ] 1 2 (α d x x 2 2) V d (2.45) To see that, define v = Ṽ 2 /2, and note that v = 2 k V x 2 2 v 0. The only two cases where v cancels are when: The solution have reached the target orbit (v = 0). The initial conditions are taken at the equilibrium x = 0. The former case shows the positive invariance of O d, the later is a consequence of the fact that the orbit should be centered at the equilibrium point (q d p, 0). 2.7 Reachability of O d Given an initial orbit (starting from given initial conditions), the system (2.1) can reach a specific periodic orbit O d through a control law for its zero-dynamics, and hence a particular dynamics of the the uncontrolled part of the system. Theorem 1. Let θ 1 = θ, θ 2 = θ, Θ = [θ 1, θ 2 ] T. Consider the following extended zero-dynamics (with φ = φ(x 1, Θ)): 4 Due to the change of coordinates x 1 = q p q d p, the orbits of the Lyapunov function V associated with the system will be centered about x 1 = 0.

37 20 CHAPTER 2. ORBITAL STABILIZATION OF UNDERACTUATED SYSTEMS: GENERAL MODEL DESCRIPTION ẋ 1 = x 2 ẋ 2 = β(x, φ, Θ, k(x, φ, Θ)) θ 1 = θ 2 θ 2 = k(x, φ, Θ) (2.46) where k(x, φ, Θ) defines the adaptation law for θ. be found such that the following statements holds: Assume that a k(x, φ, Θ) can 1. A target orbit is defined by the equation set (2.41), throughout the definition of a particular β d (x). 2. There exists a smooth function φ(x 1, Θ), and a set O Θ of suitable initial conditions for Θ(0) = Θ 0, such that: 2.1 β 2 (x 1 (t), φ(x 1 (t), Θ(t)), Θ(t)) > β(x(t), φ(x 1 (t), Θ(t)), Θ(t)) = β d (x) for all Θ(t), x 1 (t), t 0 resulting from the solution of (2.46), with x(0) 0, and Θ(0) O Θ. 3. The resulting sub-system, with x (t) = x (t + T ), and x (t) <, θ 1 = θ 2 θ 2 = k(x, φ, Θ) (2.47) yields bounded solutions. Then, for all x(0) 0 the solution x(t) converges to the target orbit.

38 2.8. CONCLUSIONS Conclusions The studied nonholonomic jointed underactuated electromechanical systems of n-dof, although usually considered to be highly nonlinear and characterized by strong nonlinear coupling, may be transformed into weakly nonlinear systems, and linear control methodology can be effectively applied. To generate stable oscillations on the system, stable limit cycles must be produced, because stable oscillations are associated to a limit cycle. To control this kind of underactuated electromechanical systems becomes much harder than in the case of holonomic systems. The studied system belongs to the class of so called nonlinear non-minimal phase systems, which still does not have satisfactory theoretical treatment.

39 22 CHAPTER 2. ORBITAL STABILIZATION OF UNDERACTUATED SYSTEMS: GENERAL MODEL DESCRIPTION

40 Chapter 3 One-Link Model Study This chapter presents an application of the control scheme proposed in the previous chapter for the orbital stabilization of an one-link underactuated system. Due to the presence of a passive joint, this system is subject to nonholonomic second-order constraints and its dynamics is more complex in comparison with full actuated systems. The control objective is to find a feedback law such that the system oscillates in a stable way. These oscillations are associated to a limit cycle. Simulation results are reported for this robot moving in a vertical plane. 3.1 One-Link Robot Model: The Reaction Wheel Pendulum In order to illustrate the control method presented in Chapter 2, we have chosen the Reaction Wheel Pendulum. This system is an underactuated electromechanical planar inverted pendulum with a symmetric rotating wheel attached to the end. This wheel has a uniform mass distribution, and it is free to spin about a parallel axis to the axis of rotation of the pendulum [SCL01]. The coupling torque is generated by a DC-motor mounted on the wheel, then the joint of the pendulum at the base is nonactuated but the wheel is actuated, see Fig

41 24 CHAPTER 3. ONE-LINK MODEL STUDY Figure 3.1: Reaction Wheel Pendulum. The Lagrangian of the underactuated system shown in Fig. 3.2 is the following: L(q, q) = 1 2 qt M q ψ(q p ) (3.1) where: q = [ qp q a ] (3.2) l : Length of link. l c : Length from the joint of wheel to the center of mass of link. r : Ratio of wheel. m 1 : Mass of link. m 2 : Mass of wheel. I 1 : Moment of inertia of link around its centroid. I 2 : Moment of inertia of wheel. q p : Angle of link (passive). q a : Angle of wheel (active). τ : Torque applied on wheel. Reaction Wheel Pendulum equations [Ol01]: m 11 q p + m 12 q a + ψ(q p ) = 0 m 21 q p + m 22 q a = τ (3.3) where: m 11 = m 1 l 2 c + m 2 l 2 + I 1 + I 2 m 12 = m 21 = m 22 = I 2 ψ(q p ) = - g (m 1 l c + m 2 l) sin(q p ) (3.4)

42 3.2. TARGET ORBITS 25 q a is a cyclic variable because doest not appear in the Lagrangian system. τ a q a r m 2 q p m 1 l c1 l 1 Figure 3.2: Reaction Wheel Pendulum system. The parameters values of this system are shown in Table 3.1 [Ol01]: Table 3.1: Parameters of the Reaction Wheel Pendulum. l = [m] l c = [m] r = [m] g = 9.8 [kg m/s 2 ] m 1 = 0.02 [kg] m 2 = 0.3 [kg] I 1 = 4.7e-5 [kg m 2 ] I 2 = 3.2e-5 [kg m 2 ] m 11 = 4.83e-3 [kg] m 12 = 3.2e-5 [kg] m 21 = 3.2e-5 [kg] m 22 = 3.2e-5 [kg] 3.2 Target Orbits In this study case, in order to reach a limit cycle that produces a stable oscillating behaviour, we consider elliptic orbits in the equation (2.41), this is: β d (x) = α d x 1 + k V x 2 Ṽ (x) (3.5) 3.3 Zero-dynamics One consequence of nonholonomy in robotic systems is that it allows to control the configuration of the whole mechanism with a reduced number of inputs.

43 26 CHAPTER 3. ONE-LINK MODEL STUDY Consider the particular case of φ(q p, θ) = a q p + b(t), θ = (a, b(t)) T with a constant, and b(t) to be adjusted as described in Chapter 2 (section 2.7). Then, the output y is: y(t) = q a (t) a q p (t) b(t) (3.6) If q d p is the desired equilibrium about which the limit cycle will be defined, then the resulting zero-dynamics expressed in the shifted coordinates is given by (2.39), with: β 2 (a) = m 1 l 2 c + m 2 l 2 + I 1 + I 2 (1 + a) (3.7) β 0 (x 1, q d p, b) = I 2 b g (m 1 l c + m 2 l) sin(x 1 + q d p) (3.8) A sufficient condition satisfying hypothesis 2.1 of the theorem 1 (section 2.7) for the existence of feasible solutions is β 2 (a) > 0, q p, is: a > (m 1 l 2 c + m 2 l 2 + I 1 + I 2 ) I 2 (3.9) 3.4 Adaptation Law The state-space representation of the zero dynamics is given by (2.40) with β = β 0 /β 2, with β 0 and β 2 as defined previously. Interestingly enough, it can be shown through direct calculations that the expression resulting from the equality 2.2 of the theorem 1 (section 2.7), is affine in b, this is: with: γ(a) b + ξ(x 1, q d p, a) = β d (x) (3.10) γ(a) = I 2 β 2 (a) ξ(x 1, q d p, a) = g (m 1 l c + m 2 l) sin(x 1 + q d p) β 2 (a) (3.11) (3.12) therefore the adaptation law k(x, φ, Θ), with b = θ 1, ḃ = θ 2, and b = k(x, φ, Θ), (a is not considered anymore as an adaptation parameter) has the form: 1 ( b = ξ(x1, q d γ(a) p, a) + β d (x) ) (3.13)

44 3.5. SIMULATIONS RESULTS 27 Note that solutions of this equation are well defined as long as γ > 0, that is if a is selected according to (3.9). This last condition can be interpreted as a physical constraint needed to generate the required oscillations. It remains to check under which conditions the assumption of the main theorem holds. In other words, under which conditions, the subsystem: θ 1 = θ 2 (3.14) θ 2 = 1 ( ξ(x γ(a) 1, qp, d a) + β d (x ) ) (3.15) has, for the specified target orbit solutions x 1(t), a bounded solution. This issue will not be studied here in detail, but we will present some simulations showing that it is indeed the case. 3.5 Simulations Results Figures show several examples of motion, when the trajectories converge to the elliptic target orbit O d (x). In order to achieve closed-loop on the system (3.3), we take any desired parameters set {V d, α d, q d p}, and its zero-dynamics is tuned through the choice of the followings parameters and initial conditions, listed in Table 3.2 and Table 3.3, respectively: Table 3.2: Parameters of the elliptic orbit. V d = 3.37e-3 [rad 2 /s 2 ] α d = 200 [1/s 2 ] q d p = 0 [rad] a = 1 k V = Table 3.3: Initial conditions of the elliptic orbits. q d p(0) = [rad] q p (0) = 0.1 [rad/s] b(0) = [rad] ḃ(0) = 0 [rad/s] Figures 3.3 and 3.4 present consecutive positions for the Reaction Wheel Pendulum for t = 0 to t = 25 sec.

45 28 CHAPTER 3. ONE-LINK MODEL STUDY T [s] Figure 3.4: Series of consecutive positions showing the oscillations in the Reaction Wheel Pendulum for t = 2.25 to t = 25 sec T [s] Figure 3.3: Sequence of simulated movements showing the oscillations in the Reaction Wheel Pendulum for t = 0 to t = 2.25 sec.

46 3.5. SIMULATIONS RESULTS 29 Figure 3.5 shows the evolution of q p when the initial position of the inverted pendulum is outside of the position of unstable equilibrium, q p (0) = - 2. This angle, measured between the vertical and the inverted pendulum, is positive clockwise. After some seconds, the system has stable oscillations around of the vertical plane q p [deg] T [s] Figure 3.5: Evolution of q p. The acting dynamics of q a is presented in Fig q a [deg] T [s] Figure 3.6: Evolution of q a. In order to compensate the initial position of the inverted pendulum, the initial speed of the link in the Reaction Wheel Pendulum is different from zero (0.1 [rad/s]). Fig. 3.7 shows the speed versus time of the wheel.

47 30 CHAPTER 3. ONE-LINK MODEL STUDY dq a /dt [rad/s] T [s] Figure 3.7: Evolution of q a. The simulation results presented in Fig. 3.8 demonstrate that the phase portrait of the Reaction Wheel Pendulum converges to the elliptic target orbit O d (x) x 2 [rad/s] (x 1 (0), x 2 (0)) O d x 1 [rad] Figure 3.8: Phase portrait of the Reaction Wheel Pendulum. Convergence to an elliptic target orbit. The zero-dynamics, showed in Fig. 3.9, converges to a stable cycle limit.

48 3.5. SIMULATIONS RESULTS db/dt [rad/s] 0 20 (b(0), db/dt(0)) b [rad] Figure 3.9: Evolution of b(t) and ḃ.

49 32 CHAPTER 3. ONE-LINK MODEL STUDY 3.6 Conclusions Several simulations of the Reaction Wheel Pendulum were performed using the control method proposed in Chapter 2. Making use of the dynamic interaction between the passive joint and active joint, the inverted pendulum has provided desirable motions. The control method to produce stable oscillations seems to be original with respect to classical biped control. These oscillations are associated to a limit cycle that is born through a Lyapunov function. Studying such a system is of significance in controlling a robot when one or more joints fail in function.

50 Chapter 4 Two-Link Model Study In this chapter we will consider the problem of stabilization of Acrobat Robot in upright position on a nonactuated horizontal bar. This underactuated electromechanical system is shown to be fully linearized and input-output decoupled by means of a nonlinear dynamic feedback. Simulation results are reported for this robot moving in a vertical plane. 4.1 Two-Link Robot Model: The Acrobat Robot The Acrobot, short for ACRObat robot, is an electro-mechanical system consisting of two rigid links interconnected by revolute joints. The Acrobot is termed underactuated, as it contains fewer actuators than degrees of freedom [FLS00]. The first joint is nonactuated and the second joint is actuated by a DC-motor. This robot is nonholonomic because its acceleration constraint cannot be integrable. The robot used in this section, named Super Mechano-Boy, is anthropomorphic, then it is more complex than a typical Acrobot. It has 9 links: arms, body, legs, tibia, and feet. If the Super Mechano-Boy is mounted so that the joint axes are perpendicular to gravity, then there will be a continuum of equilibrium configurations, each corresponding to a constant value of the input torque. In this work the Super Mechano-Boy is mounted so that the joint axes are parallel to gravity, where we consider the standard assumption, i.e., the robot links are considered to be rigid and their friction is rejected. In our study, we consider a 2 link model (arms and body) because we freeze the other links; it is shown in the Fig The arms are connected and rotates freely with a horizontal bar and the bar is not actuated. 33

51 34 CHAPTER 4. TWO-LINK MODEL STUDY Figure 4.1: Acrobot. Super Mechano-Boy Consider the underactuated planar robot called Acrobot, shown schematically in Fig m 2 τ a q a l 2 l c2 m 1 q p l c1 l 1 Figure 4.2: Acrobot system. l i : Length of link i. l ci : Length from the previous joint to the center of mass of link i. m i : Mass ok link i. I i : Moment of inertia of link i around its centroid. q p : Angle of link passive. q a : Angle of link active. τ a : Torque applied on link 2. i = 1, 2. Acrobot equations [Bo97]: [ ] ( ) Mp (q) m11 m M(q) = = 12 M a (q) m 21 m 22 and N(q, q) = [ Na N p ] ( ) [ ] C11 C = 12 qp + C 21 C 22 q a [ g1 g 2 ] (4.1) (4.2)

52 4.2. TARGET ORBITS 35 where: m 11 = m 1 l 2 c1 + m 2 (l 2 c2 + l l 1 l c2 cos(q a )) + I 1 + I 2 m 12 = m 21 = m 2 (l 2 c2 + l 1 l c2 cos(q a )) + I 2 m 22 = m 2 l 2 c2 + I 2 C 11 = - m 2 l 1 l c2 sin(q a ) q a C 12 = - m 2 l 1 l c2 sin(q a ) ( q p + q a ) C 21 = m 2 l 1 l c2 sin(q a ) q p C 22 = 0 g 1 = g ((m 1 l c1 + m 2 l 1 ) cos(q p ) + m 2 l c2 cos(q p + q a )) g 2 = g m 2 l c2 cos(q p + q a ) (4.3) The parameters values are shown in Table 4.1 [YYML02]: Table 4.1: Parameters of the Acrobot. l 1 = 0.18 [m] l c1 = 0.11 [m] l c2 = 0.16 [m] g = 9.8 [kg m/s 2 ] m 1 = 0.52 [kg] m 2 = 3.17 [kg] I 1 = 1.4e-3 [kg m 2 ] I 2 = 4.4e-2 [kg m 2 ] 4.2 Target Orbits For this study case, we consider elliptic orbits in the equation (2.41) i.e., β d (x) = α d x 1 + k V x 2 Ṽ (x) (4.4) 4.3 Zero-dynamics Consider the particular case of φ(q p, θ) = a q p + b(t), θ = (a, b(t)) T with a constant, and b(t) to be adjusted as described previously. Then, the output y is: y(t) = q a (t) a q p (t) b(t) (4.5) If q d p is the desired equilibrium around what the limit cycle will be defined, then the resulting zero dynamics expressed in the shifted coordinates is given by (2.39), with: β 2 (x 1, qp, d a, b) = m 1 lc1 2 + m 2 (l 1 l c2 cos(a (x 1 + qp) d + b)) (2 + a) + lc2 2 (1 + a) + l1) 2 + I 1 + I 2 (1 + a) (4.6) β 0 (x, qp, d a, b, ḃ, b) = σ 2 (x 1, qp, d a, b) x σ 1 (x 1, qp, d a, b, ḃ) x 2 + σ 0 (x 1, qp, d a, b, ḃ, b) (4.7)

53 36 CHAPTER 4. TWO-LINK MODEL STUDY where: σ 2 (x 1, q d p, a, b) = a m 2 l 1 l c2 sin(a (x 1 + q d p) + b) (2 + a) (4.8) σ 1 (x 1, q d p, a, b, ḃ) = 2 m 2 l 1 l c2 sin(a (x 1 + q d p) + b) (1 + a) ḃ (4.9) σ 0 (x 1, q d p, a, b, ḃ, b) = (m 2 (l 2 c2 + l 1 l c2 cos(a (x 1 + q d p) + b)) + I 2 ) b m 2 l 1 l c2 sin(a (x 1 + q d p) + b) ḃ2 + g ((m 1 l c1 + m 2 l 1 ) cos(x 1 + q d p) + m 2 l c2 cos(1 + a) ((x 1 + q d p) + b)) (4.10) A sufficient condition satisfying hypothesis 2.1 of the theorem 1 (section 2.7) for the existence of feasible solutions is β 2 = m 22 + a m 12 > 0, q p. 4.4 Adaptation Law The state-space representation of the zero dynamics is given by (2.40) with β = β 0 /β 2, with β 0 and β 2 as defined previously. Interestingly enough, it can be shown by direct calculations that the expression resulting from the equality 2.2 of the main theorem 1 (section 2.7), is affine in b, this is: γ(x 1, q d p, a, b) b + ξ(x, q d p, a, b, ḃ) = β d(x) (4.11) with: γ = m 2 (l 2 c2 + l 1 l c2 cos(a (x 1 + q d p) + b)) + I 2 β 2 (x 1, q d p, a, b) ξ = σ 2 x σ 1 x 2 + ζ β 2 (x 1, q d p, a, b) (4.12) (4.13) ζ = m 2 l 1 l c2 sin(a (x 1 + q d p) + b) ḃ2 + g ((m 1 l c1 + m 2 l 1 ) cos(x 1 + q d p) + m 2 l c2 cos(1 + a) ((x 1 + q d p) + b)) (4.14) therefore the adaptation law k(x, φ, Θ), with b = θ 1, ḃ = θ 2, and b = k(x, φ, Θ), (a is not considered anymore as an adaptation parameter) has the form: 1 ( ) b = ξ(x, q d γ(x 1, qp, d a, b) p, a, b, ḃ) + β d(x) (4.15) Note that, solutions of this equation are well defined as long as γ > 0, that is if a is selected according to (3.9). This last condition can be interpreted as a physical constraint

54 4.5. SIMULATIONS RESULTS 37 needed to generate the required oscillations. It remains to check under which conditions the assumption of the main theorem holds. In other words, under which conditions, the subsystem: θ 1 = θ 2 (4.16) θ 2 = 1 ( ) ξ(x, q d γ(x 1, qp, d a, b) p, a, b, ḃ) + β d(x ) (4.17) has, for the specified target orbit solutions x (t), a bounded solution. This issue, will not be studied here in detail, but we will only present some simulations showing that it is indeed the case. 4.5 Simulations Results Figures show several examples of motion, where the trajectories converge to the elliptic orbit O d (x). For achieving this elliptic orbit, we can take any desired parameters set {V d, α d, qp}, d from Table 4.2 and Table 4.3: Table 4.2: Parameters of the elliptic orbits. V d = 1.5e-4 [rad 2 /s 2 ] α d = [1/s 2 ] q d p = π/2 [rad] a = 4 k V = 2500 Table 4.3: Initial conditions of the elliptic orbits. q d p(0) = [rad] q p (0) = [rad/s] b(0) = [rad] ḃ(0) = 0 [rad/s] In Fig. 4.3, we present a series of consecutive positions of the Acrobat Robot showing that the oscillations around its upright equilibrium point are achieved T [s] Figure 4.3: Sequence of images showing the oscillations in the Acrobot for t = 0 to t = 25 sec.

55 38 CHAPTER 4. TWO-LINK MODEL STUDY From Fig. 4.4 we can see that the underactuated joint q p has an oscillatory behaviour imposed by the actuated joint q a (Figure 4.5). The first joint corresponding to the hand on the bar cannot exert torque, but the second joint, corresponding to the gymnast bending at the waist, does it q p [deg] T [s] Figure 4.4: Evolution of q p (t). Chaotic behaviour can be appreciated in the position q a (t), and speed q a, see Figs. 4.5 and q a [deg] T [s] Figure 4.5: Evolution of q a (t).

56 4.5. SIMULATIONS RESULTS dq a /dt [rad/s] T [s] Figure 4.6: Evolution of q a. The zero-dynamics also shows chaotic behaviour, see Figs. 4.7 and b [deg] T [s] Figure 4.7: Evolution of b.

57 40 CHAPTER 4. TWO-LINK MODEL STUDY db/dt [rad/s] T [s] Figure 4.8: Evolution of ḃ. The simulation results, see Fig. 4.9, show that the system converges to the orbit O d (x) starting from initial conditions outside of the stable limit cycle (elliptic target orbit) x 2 [rad/s] (x 1 (0), x 2 (0)) O d x 1 [rad] Figure 4.9: Convergence to an elliptic target orbit O d (x). The zero-dynamics of this system is bounded, as showed in Fig

58 4.5. SIMULATIONS RESULTS db/dt [rad/s] b [rad] T [s] Figure 4.10: Evolution of b(t) and ḃ.

59 42 CHAPTER 4. TWO-LINK MODEL STUDY 4.6 Conclusions A 2-DOF system was controlled in a standing and not tracking position. The controller should keep the robot s center of mass directly above the pivot point for stability. Simulation results were provided for parameters taken from a real-life model of the Acrobat Robot. These results are satisfactory since in spite of the non-actuated joint i.e., when only q a can be controlled directly, the dynamic system showed orbital stabilization.

60 Chapter 5 Tree-Link Model Study This chapter presents an application of the control scheme proposed in the previous chapters of a bipedal robot in the swing phase. It shows how to reduce the stability analysis to a one-degree of freedom dynamical system, and gives general conditions to ensure oscillatory behaviour with internal stability. Since a bipedal robot can be represented by an inverted pendulum system, we deal with the orbital stabilization problem of a 3-DOF inverted pendulum. In this high-dimensional inverted pendulum, the first link has not direct control input and the contact with the ground is modeled as a pivot. The control objective aims at finding a feedback law such that the inverted pendulum be able to reach a periodic motion in a stable way. We show that changing the period and/or the amplitude of the motion, we can have jumps from one admissible limit cycle to another 1. To generate an oscillatory behaviour in whole underactuated inverted pendulum coordinate set, the methodology presented in Chapter 2-4 (assume that the suitable m-dimensional output has the form given by Eq. (2.8) (y(q, t) := q a φ(q p, θ(t))), with q = [ ] qp T, qa T T ), may be summarized in the following phases: 1. Choosing a suitable output function. 2. Linearizing such output. 3. Studying the conditions under which this output (y(q, t)) may create a stable limit cycle in the resulting zero-dynamics. 4. Applying feedback stabilization to the inverted equilibrium manifold. As example, we can take a linear constraint of the form: q a = a q p + b(t) (5.1) where a R (m, n m) = R (m, 1) is a constant vector, b(t) R m is a vector allowed to vary with the time. In particular, if a = (a 1, a 2,... a m ) T, then the vector: θ(t) = [a T, b(t) T ] T (5.2) 1 The admissible limit cycles are not completely free because they must respect the constraints imposed by the physics of the workspace and contact forces at the leg end. These restrictions disqualify feedbacks leading to the natural homoclinic orbits (orbits with constant energy levels), that may induce inadmissible motions out of the joint workspace. 43

61 44 CHAPTER 5. TREE-LINK MODEL STUDY characterizes the parameters to be tuned (or adapted). 5.1 Three-Link Robot Model: The Gymnastic Robot The considered example is an underactuated three-link planar robot depicted in Fig. 5.1, which is free to swing in the vertical plane. Consider the three-link planar robot equations [ATS98]: m 11 q p + m 12 q a1 + m 13 q a2 + c 11 q p + c 12 q a1 + c 13 q a2 + g 1 = 0 (5.3) m 21 q p + m 22 q a1 + m 23 q a2 + c 21 q p + c 22 q a1 + c 23 q a2 + g 2 = τ a1 (5.4) m 31 q p + m 32 q a1 + m 33 q a2 + c 31 q p + c 32 q a1 + c 33 q a2 + g 3 = τ a2 (5.5) where the exact expressions for m ij (q a1, q a2 ), c ij (q a1, q a2, q p, q a1, q a2 ), and g i (q p, q a1, q a2 ), i, j = 1,...3. were especially developed for this thesis, and can be found in Appendix D. The subscripts a1 and a2 are used to represent quantities relative to the active joints, whereas p represents this for the passive joint. Eq. (5.3) represents a dynamics constraint witch cannot be integrated to obtain an algebraic relation between the coordinates q p, q a1, and q a2. - τ a2 - q a2 m 3 l c3 l 3 m 2 τ a1 l 2 l c2 q a1 m 1 q p l c1 l 1 Figure 5.1: Three-link planar robot. The following physical parameters for the robot, assumed for the calculations, are shown in Table 5.1:

62 5.2. ZERO-DYNAMICS 45 Table 5.1: Parameters of the Gymnastic robot. l 1 = 0.47 [m] l c1 = 0.14 [m] m 1 = 1.48 [kg] I 1 = [kg m/s 2 ] l 2 = 0.26 [m] l c2 = 0.12 [m] m 2 = 1.06 [kg] I 2 = [kg m/s 2 ] l 3 = 0.29 [m] l c3 = 0.25 [m] m 3 = 0.29 [kg] I 3 = [kg m/s 2 ] The system (5.3)-(5.5) projected on the constraint y = ẏ = ÿ = 0, gives us the zero-dynamics. The resulting zero-dynamics can be computed by substituting these constraints in the unactuated part of the system. By means of the choice of a class of output functions, we can parameterize the zerodynamics. For the system (5.3)-(5.5), we define the following 2-dimensional output: [ ] qa1 a y(q, t) := 1 q p b 1 (t) (5.6) q a2 a 2 q p b 2 (t) 5.2 Zero-dynamics Complexity and dimension of the zero-dynamics increases if both b i (t) parameters are let to be time varying. For simplicity of the design, only b 1 (t) is let to be varying while b 2 as well as a 1, and a 2 are set to constant values. The parameter vector is thus defined as θ = [a 1, a 2, b 1 (t), b 2 ] T. This particular choice leads to the following constraints: [ ] [ ] qa1 a1 q = p + b 1 (t) (5.7) q a2 a 2 q p + b 2 [ ] [ qa1 a1 q = p + ḃ1(t) ] (5.8) q a2 a 2 q p [ ] [ qa1 a1 q = p + b ] 1 (t) (5.9) q a2 a 2 q p and substituting in (5.3), the resultant zero-dynamics has the form: β 2 (q p, φ, θ) q p + β 0 (q p, q p, φ, θ, θ, θ) = 0 (5.10) whose solution will be well defined if β 2 (q p, θ) > 0. For notation simplicity we will drop the time argument in what follows. 5.3 Forced Zero-dynamics The method introduced in Chapter 2 consists in finding a control law that matches the open loop system with that desired behaviour. Following the development presented

63 46 CHAPTER 5. TREE-LINK MODEL STUDY in Chapter 2-4, the zero-dynamics (5.10) can be forced to match a stable cyclic motion. This can be done as follows: define first x 1 = q p q d p and x 2 = q p, with q d p being the center of the desired orbit, and from (5.10), we have: ẋ 1 = x 2 ẋ 2 = β(x, φ, θ, θ, θ) (5.11) Considering the desired family of elliptic orbits given by V d = 1 2 (α dx x 2 2), characterized by the design parameters, the zero-dynamics (5.11) is forced to behave as the orbit generator (2.41) if we can find an adaptation law for θ, such that the following differential equation is satisfied: β(x, φ, θ, θ, θ) = β d (x), t 0 (5.12) Let θ 1 = b 1, θ 2 = θ 1, then the adaptation law takes, in our particular case, the form: θ 1 = θ 2 1 θ 2 = (f m 12 (x 1,θ) 5(x 1, θ) θ2 2 + f 7 (x 1, θ) x 2 θ 2 + f 8 (x 1, θ) x g 1 (x 1, θ) + β 2 (x 1, θ) β d (x)) (5.13) A suitable choice of a is necessary in order to get m 12 > 0. However, due to the quadratic terms in θ, initial conditions have to be properly selected as to avoid finite escape time. We have also seen that there exists a compact set of initial conditions such that the equation (2.46) remain asymptotically bounded. 5.4 Simulations Results We programmed the robot to show stability and periodicity swingup control. Figures show an example of motion, where the trajectory converge to the orbit O d. The chosen parameters values are listed in Table 5.2, and the initial conditions are shown in Table 5.3: Table 5.2: Parameters of the elliptic orbits. V d = [rad 2 /s 2 ] α d = [1/s 2 ] a 1 = 1 a 2 = 2 b 2 = [rad] q d p = π/2 [rad] k V = 100 Table 5.3: Initial conditions of the elliptic orbits. q d p(0) = [rad] q p (0) = 0.3 [rad/s] b 1 (0) = [rad] ḃ 1 (0) = 0 [rad/s]

64 5.4. SIMULATIONS RESULTS 47 Figure 5.2 shows the evolution of q p where the initial position of the Gymnastic Robot is outside of the position of unstable equilibrium, q p (0) = This angle, measured between the horizontal and the first link of the inverted pendulum, is negative clockwise. After some seconds, the system has stable oscillations around the vertical plane q p [deg] T [s] Figure 5.2: Evolution of q p. The acting dynamics of q a1 and q a2 is presented in Figs. 5.3 and q a1 [deg] T [s] Figure 5.3: Evolution of q a1 (t). In order to compensate the initial position of this 3-DOF inverted pendulum, the initial speed of the non actuated link is different from zero (0.3 [rad/s]). Figs. 5.5 and 5.6 show the speed versus time of the actuated links.

65 48 CHAPTER 5. TREE-LINK MODEL STUDY q a2 [deg] T [s] Figure 5.4: Evolution of q a2 (t) dq a1 /dt [rad/s] T [s] Figure 5.5: Evolution of q a1.

66 5.4. SIMULATIONS RESULTS dq a2 /dt [rad/s] T [s] Figure 5.6: Evolution of q a2. The simulation results presented in Fig. 5.7 demonstrate that the phase portrait of the three-link model study converges to the elliptic target orbit O d (x) (x 1 (0), x 2 (0)) 0.2 x 2 [rad/s] O d x [rad] 1 Figure 5.7: Phase portrait of the Gymnastic Robot. Convergence to an elliptic target orbit. The zero-dynamics, showed in Figs. 5.8 and 5.9, converges to a stable cycle limit.

67 50 CHAPTER 5. TREE-LINK MODEL STUDY b 1 [deg] T [s] Figure 5.8: Evolution of b 1 (t) db 1 /dt [rad/s] T [s] Figure 5.9: Evolution of ḃ1. The zero-dynamics of this system is bounded, this is showed in Fig

68 5.4. SIMULATIONS RESULTS db 1 /dt [rad/s] b 1 [rad] T [s] Figure 5.10: Evolution of b 1 (t) and ḃ1.

69 52 CHAPTER 5. TREE-LINK MODEL STUDY 5.5 Conclusions This chapter addressed the orbital stabilization problem of a three-link planar robot. We showed that the orbital stability of such a system can be accomplished in spite of using one less actuator than DOF s to adjust the center of mass. The obtained results give the possibility of having an adaptation law of higher dimension, i.e., to work with systems of higher dimension using the same methodology used in the Gymnastic Robot. However, this additional degree of freedom does not provide further possibility to enhance the system stability property. Nevertheless, it results in a control structure providing similar stability properties in spite of the increase of system complexity. Therefore, this study also shown that it is possible to treat cases where the dimension of the robot is even larger. The motivation of the research reported in this chapter is to extend this result to high-dimensional systems (walking mechanisms). In the following chapter, we will present some of the results obtained by applying this control strategy on our dynamic model, a 5-DOF walking mechanism.

70 Chapter 6 Five-Link Model Study: The RABBIT Robot This chapter is devoted to present the extension of the proposed methodology 1, shown in the previous chapters, to a five-link model, the Rabbit Robot. This is a natural continuation of the approach presented in [CEU02, UCM02]. To enable testing and analyzing of Rabbit robot s behaviour using this control strategy, we use a graphical simulation program, nicked LAPIN. We modified this simulator to allow virtual experiments including our control system. LAPIN was used to display the movement of links, the desired and actual positions, the torque requests, etc. 6.1 LAPIN: The Rabbit s Dynamic Model Simulator For validate our theoretical work (before implementing our experiments), we use the LAPIN simulator. LAPIN is a simulator of a 2D underactuated bipedal robot with five rigid limbs, as the Rabbit robot. The kinematic architecture of the robot retained is a five links system (trunk, two femurs, and two tibias), and four revolute actuated joint, see Fig An approach to control of bipedal robots. 53

71 54 CHAPTER 6. FIVE-LINK MODEL STUDY: THE RABBIT ROBOT Z q T z q RK -q RH q LH -q LK Y x X Figure 6.1: Rabbit s model representation. with: q T : Angle of trunk. q RH : Angle of right hip. q RK : Angle of right knee. q LH : Angle of left hip. q LK : Angle of left knee. LAPIN is a common platform of experiments for several french laboratories 2 in order to make comparisons between different control laws. It uses an approximate dynamic model for a five-link planar robot, and it allows every of the design parameters to be changed. This simulator improves the knowledge on the relation between the mechanical part and the control part of a bipedal robot, in order to obtain cyclic motions with a low energy consumption. 2 LAG, INRIA Rône-Alpes, INRIA Sophia-Antipolis, LMS, LSIIT-GRAVIR, IRCyN, and LRP, joint together in the CNRS Project Control of Legged Robots.

72 6.1. LAPIN: THE RABBIT S DYNAMIC MODEL SIMULATOR Basic Features Graphical interface Parametrization It includes a contact model between the foot and the floor Capacity to output the results easily Easy access to the parameters Structure The chosen development platform is Matlab 5.1/Simulink 2.1 software of the Math- Works Inc. A main specification point is the possibility to change any parameter (geometric, masses, simulation conditions, ground parameters, initial conditions), except the kinematic architecture, and the capacity to held chosen parameters by default. LAPIN s structure is represented in Fig Three main uses are possible. The first one is the performance evaluation of a structure 3 towards initial conditions or external constraints 4, this way is represented by the Fig. 6.2 central column. The second is the control laws implementation (Fig. 6.2 right side). This use is reserved for control design or to adjust the gains. The last one represents the prototyping evaluation use i.e., for adjusting the mass distribution or segment length. Initial condition and physical parameters Mechanical and system parameters for prototyping evaluation Dynamic simulation Control laws implementation Results presentation Graphical animation Curves Figure 6.2: Block diagram for the LAPIN s structure. 3 A defined control scheme for a specific biped. 4 Variation of grounds parameters.

73 56 CHAPTER 6. FIVE-LINK MODEL STUDY: THE RABBIT ROBOT 6.2 Transformation to Three-DOF Underactuated Mechanisms To apply the methodology described in the last chapters (on Rabbit robot), it is necessary to carry out a transformation that allows to work with this robot as if Rabbit had three degrees of freedom. For doing this, we freeze the robot s right knee and we calculate the position of the center of equivalent mass of the trunk-femur-tibia group. This group will have a single degree of freedom (see Fig. 6.3c) that, together with the degree of freedom of the trunk/left-femur and with the degree of freedom of the left-femur/left-tibia, give us a 3DOF system, then: Z ϕ 1 m 4 l 4 m 4 Z ϕ 3 ϕ 4 = 0 l 1 X l 3 X l 5 m 5 l 2 Z' ϕ 2 m 1 m 3 X ' m 2 (a) (b) (c) Figure 6.3: Rabbit robot system: (a) 5DOF, (b) 4DOF, and (c) 3DOF. l i+2 = m i l i cos(ϕ i ) + m i+1 l i+1 cos(ϕ i+1 ) m i+2 cos(ϕ i+2 ) (6.1) Then for i = 1, 3. l 3 = [m], l 5 = [m], ϕ 3 = 180 [deg] ϕ 5 = 180 [deg] We assume that the plane of motion for the robot is tilted by a generic angle ϕ w.r.t. the vertical plane through the x-axis.

74 6.3. SIMULATIONS RESULTS Simulations Results In this section we present a series of simulations whit the objective of driving both active and passive joints of the robot from an initial configuration to a given orbit O d. In this strategy the required leg joint moments were estimated by an orbit generator. The parameters of the robot are shown in Table 6.1, the parameters of the elliptic orbits are listed in Table 6.2, and the initial conditions are shown in Table 6.3: Table 6.1: Parameters of the robot. l 1 = 0.8 [m] l c1 = [m] m 1 = 22 [kg] I 1 = 4.23 [kg m/s 2 ] l 2 = 0.4 [m] l c2 = [m] m 2 = 6.8 [kg] I 2 = 1.08 [kg m/s 2 ] l 3 = 0.4 [m] l c3 = [m] m 3 = 3.2 [kg] I 3 = 0.93 [kg m/s 2 ] Table 6.2: Parameters of the elliptic orbits. V d = [rad 2 /s 2 ] α d = [1/s 2 ] a 1 = 2 a 2 = 10 b 2 = [rad] qp d = π/2 [rad] k V = 1 Table 6.3: Initial conditions of the elliptic orbits. q d p(0) = [rad] q p (0) = 0 [rad/s] b 1 (0) = 0 [rad] ḃ 1 (0) = 0 [rad/s] The first simulation consisted in driving the support-leg/ground joint (non-actuated joint) over a 90 excursion. This angle, measured between the horizontal and the leg, is negative clockwise. Figure 6.4 presents the results of control T [s] Figure 6.4: Series of consecutive positions showing the oscillations in the Rabbit s model for t = 0 to t = 25 sec. Figures show several examples of motion, where the trajectories converge to the orbit O d.

75 58 CHAPTER 6. FIVE-LINK MODEL STUDY: THE RABBIT ROBOT q RH [deg] T [s] Figure 6.5: Evolution of q RH (t) q RK [deg] T [s] Figure 6.6: Evolution of q RK (t).

76 6.3. SIMULATIONS RESULTS q LH [deg] T [s] Figure 6.7: Evolution of q LH (t) q LK [deg] T [s] Figure 6.8: Evolution of q LK (t).

77 60 CHAPTER 6. FIVE-LINK MODEL STUDY: THE RABBIT ROBOT dq LH /dt [deg/s] T [s] Figure 6.9: Evolution of q LH (t) b 1 [deg] T [s] Figure 6.10: Evolution of b 1 (t).

78 6.3. SIMULATIONS RESULTS db 1 /dt [deg/s] T [s] Figure 6.11: Evolution of ḃ x 2 [rad/s] x 1 [rad] Figure 6.12: Convergence to an elliptic target orbit O d (x).

79 62 CHAPTER 6. FIVE-LINK MODEL STUDY: THE RABBIT ROBOT db 1 /dt [rad/s] b 1 [rad] T [s] Figure 6.13: Evolution of b(t) and ḃ1.

80 6.4. CONCLUSIONS Conclusions In this chapter we have proved a control system capable of generating stable balancing in a bipedal robot. To accomplish these oscillations, an appropriate control law that drives the system to a stable limit cycle has been introduced. For successful balance, the base of support must remain, on average, under the centre of mass and the trunk also should maintain, on average, an erect posture. The key characteristic of this strategy compared to many other closed-loop strategies proposed in the literature is that no reference information, such as desired trajectory or path, is required.

81 64 CHAPTER 6. FIVE-LINK MODEL STUDY: THE RABBIT ROBOT

82 Chapter 7 Experimental Results on RABBIT Robot This chapter presents a brief description of the Rabbit s hardware and software. The robot s dynamics is transformed into a fully linear, input-output decoupled system by using a second-order dynamic feedback compensator. 7.1 The Rabbit Robot Description Rabbit is a planar robot that was built for research purposes. This testbed allows to validate, experimentally, theoretical tools and control methods to generate closed-loop motions in bipedal robots. Rabbit allows us to study interesting problems such as periodic and stable limit cycles in balancing phases (i.e., the robot is dynamically standing on one leg), walking and running. The design philosophy of Rabbit was to follow an anthropomorphic design and at the same time keep everything as simple as possible, i.e., still being representative of human walking [CL02]. The mechanism has seven DOF: five rigid links connected to one another by purely rotational joint angles plus two DOF associated with the horizontal and vertical displacement of the center of mass 1 within the sagittal plane. Rabbit has a trunk and two legs with knees but no feet. The axes between the trunk and each femur are actuated, as the axes of each knee, but the trunk is non actuated. So, this prototype has seven degrees of freedom and four degrees of actuation. The contact between the support leg and ground is modeled as a pivot, then the contact between leg and soil is prompt, and the model has five degree of freedom. The support-leg/ground joint is unilateral since attractive forces are absent, and passive because it is underactuated, thus Rabbit cannot apply torques to the ground. To guarantee lateral stability, there is an external support structure that does not allow motions in the transverse or coronal planes, see Fig In this way, Rabbit s motion is limited to an evolution on the sagittal plane, maintained by a stem around a central post by means of a radial bar 2, see Fig The robot is free in rotation around 1 The Cartesian coordinates of the hips. 2 The radius of the circular path is approximately 3 m. 65

83 66 CHAPTER 7. EXPERIMENTAL RESULTS ON RABBIT ROBOT the stem, and it has rubber wheels located under the heels 3. These wheels, on its coronal plane, are used to permit frictionless radial displacement of the supporting leg. Transverse Coronal (frontal) Sagittal Figure 7.1: Reference planes of the human body. There are 3-DOF between Rabbit and the central column, a revolute joint between the trunk and the radial bar that is aligned with the axes of the hips (rotational motion), plus a universal joint between the radial bar and the central column (horizontal and vertical motion). In this way, the three passive DOF of the biped on the sagittal plane are allowed to give the robot complete mobility in this plane. 3 The wheels were constructed of a stiff polymer for shock absorbing.

84 7.2. HARDWARE DESCRIPTION 67 Figure 7.2: Rabbit s central column and circular path. 7.2 Hardware Description The prototype stands m high and weighs approximately 32 kg. Femur mass is 6.8 kg (including half-reducer hip, motor knee and half-reducing knee), Tibia mass is 3.2 kg (including half-reducing knee and tibia), Trunk mass is approximately 12 kg. Its geometry is detailed in Fig The main physical parameters on Rabbit are shown in Table 7.1, and the Rabbit s inertia reducer parameters are listed in Table 7.2: Table 7.1: Main physical parameters on Rabbit. Trunk Femur Tibia Length [m] M ass [kg] Center of mass [m] yg = 0.01; zg = 0.2 zg = zg = Inertia [kg m 2 ] Iox = 2.22 Iox = 0.25 Iox = 0.10 F irst moments [kg m] MY = 0.2; MZ = 4.0 MZ = 1.11 MZ = 0.41 Corrected inertia [kg m 2 ] Iox = 1.08 Iox = 0.93 Table 7.2: Rabbit s inertia reducer parameters. Irotor [kg m 2 ] Ireducer [kg m 2 ] Irotor + Ireducer [kg m 2 ] 2.25e e e-4 In Rabbit, aluminium was chosen as frame material, because it provides high strength with a relatively low weight. All mechanical and electrical components are mounted as

85 68 CHAPTER 7. EXPERIMENTAL RESULTS ON RABBIT ROBOT Trunk Motor and incremental encoder (hip) Gear reducer and absolute encoder (hip) Motor and incremental encoder (knee) Gear reducer and absolute encoder (knee) Figure 7.3: Rabbit s robot geometry. symmetrically as possible in order to keep the center of mass along lines of symmetry in the robot. In order to minimize the amount of unnecessary movements in all joints, each linkage is designed to provide the necessary range of motion while maintaining a tight connection, see Table 7.3. This is accomplished differently for each joint. Table 7.3: Association between the sign of the control torque/degrees and the action of motors. Joint Degree (backward) Degree (f orward) Bar/central column (hip) around the horizontal Limitless rotation Limitless rotation axis Bar/central column (hip) around the vertical axis Bar/trunk (hip) T runk/f emur (hip) - 80 (extension) 80 (f lexion) F emur/tibia (knee) - 50 (f lexion) 50 (extension) Actuators. The four actuated joints in Rabbit are motorized by Parvex RS420J motors with Samarium Cobalt magnets (4 N m at 2000 r.p.m., 1 N m at 5000 r.p.m., mass: 2.2 kg.). These actuators are powered by an external 60 V DC source, isolated from the computing power supply. The motors are associated to gear reducers Harmonic-Drive (ratio: 1/50, mass: 2.1 kg.) and belts to activate the four joints. All motors and gear

86 7.3. SOFTWARE DESCRIPTION 69 reducers are positioned as high as possible above their corresponding joints in order to reduce the mass on the lower extremities, and hence decreasing the load on a given actuator 4. The actuators use standard model airplane connectors on their shafts. Sensors. As in all actuated systems, the sensors facilitate the feedback for position control. Rabbit is a planar robot, then a two-axis sensor is required. Each actuated axis is equipped with two sensors to measure angular position (the velocity is calculated from its position). Identical sensors are used at each actuated joint; one sensor is attached directly to the motor shaft, while the second one is attached to the shaft of the gear reducer. Every actuated axe is furthermore provided of two sensors of end of race to every extremity of the race. These sensors possesses two contacts (1O+1C). The motor position sensors are incremental encoders with 8192 counts/revolution. The sensors mounted on the gear reducers Harmonic-Drive are absolute encoders with 250 counts/revolution. The absolute encoders measure directly the relative angles between the links, while the relative encoders yield an indirect measure of the relative link angles due to the presence of a gear-reducer and belt. The mechanism has three additional encoders. One measures the absolute angle of the trunk with respect to the stabilizing bar. The second measures the horizontal angle of the stabilizing bar with respect to the central column; this allows the distance travelled by the robot to be computed. The final sensor measures the vertical angle of the stabilizing bar, which allows the height of the hips to be measured; in single or double support, this information is redundant, but during the flight phase, it is not. The support leg and double support phases are therefore easily distinguished. In the implementation of our experiments we used the real-time operational system DSpace. Our control software runs under this multitasking multiprocessor operating system, designed to support software written as a set of modules running independently. DSpace allows us easy prototyping of controllers within the Simulink environment and downloading these controllers to dedicated Digital Signal Processor based systems for implementation. 7.3 Software Description The control software was written in Matlab 5.3/Simulink 3.0, and our real-time graphical interface implemented in ControlDesk by Gabriel Buche 5 and Claudio Urrea. ControlDesk is the integrated tool for the control, monitoring, and automation of our real-time experiments. It allows us real-time data acquisition, online parameter tuning, and perfect connection to Simulink and SystemBuild. A software controller was implemented for Rabbit. This software controller uses a simple local PID control algorithm, and simulates four decoupled PID controllers, one 4 Specifically, to minimize the inertia of the legs, the motors for the knees are mounted close to hips. 5 Electric engineer.

87 70 CHAPTER 7. EXPERIMENTAL RESULTS ON RABBIT ROBOT for each actuated link. This control algorithm give us simplicity in the design, and to know a precise model of robot is not necessary. The low-level part of motor behaviours which takes care of the torque requests for individual motors can be implemented very well using standard PID regulators. Thus the robot is driven by four independent PID controllers and friction is compensated by our control software. For the controllers design, the values of the dynamic parameters of PID controllers were selected knowing the demand of each w n (bandwidth). By employing the FFT (Fast Fourier Transformation) on the trajectories generated by our orbital controller, we can obtain the w n of every actuated joint. Figure 7.4 and Table 7.4 summarize the results.

88 7.3. SOFTWARE DESCRIPTION A q RH A q RK ω [rad/s] ω [rad/s] (a) (b) A q LH 10 2 A q LK ω [rad/s] ω [rad/s] (c) (d) Figure 7.4: FFT on angular position in joints: (a) Right hip, (b) Right knee, (c) Left hip, and (d) Left knee.

89 72 CHAPTER 7. EXPERIMENTAL RESULTS ON RABBIT ROBOT Table 7.4: Values of the dynamic parameters of PID controllers. Joint Proportional Integral Derivative Hip lef t Knee lef t Hip right Knee right Inner-Loop It is necessary to understand that from automatic control point of view, there are important difficulties to control and to stabilize this robot [CAAPWCG02]: As the system is underactuated, we cannot hope to controller all the DOF on the previously defined trajectories 6. The contact between the stance leg and the ground is virtually pointwise and can be broken in presence of disturbances 7. A model that captures all the main physical phenomena is mathematically very complex. There are many sources of error between the idealized Lagrangian dynamics and the actual system: Friction in the motor-belt-gear reducer system that is used at each joint, as well as friction at the universal joint of the tower that supports the power electronics and the DSpace system. Unmodeled flex dynamics caused by cabling, torsion of the gears in the joints, and flexing of the counter-balance bar. Parameter inaccuracies due to poor estimates of link inertias and the additional inertia of the tower that contains the DSpace module and power electronics. Non-rigid impacts due to compliance at the end of the leg and in the walking surface. Digital implementation issues such as sampling effects, quantization, velocity estimation from position measurements. The classic tools of the automatic control for the Rabbit s stability analysis are inefficient, then the concepts of stability must be correctly adapted to the robot. 6 In our work, we drive both active and passive joints of the robot from an initial configuration to a target orbit O d. 7 We assume that there is sufficient friction between the feet and the ground surface to prevent slippage.

90 7.5. COMMENTS ON EXPERIMENTAL RESULTS 73 Perturbations d d d d (q RH, q LH, q RK, q LK ) + _ Controller RABBIT Robot + _ q = [q T, q RH, q LH, q RK, q LK ] (q RH, q LH, q RK, q LK ) Orbits Generator q q T d Rabbit's parameters V d αd K a 1 a 2 b 1 (0) b 2 Figure 7.5: Rabbit Block Control. 7.5 Comments on Experimental Results In order to evaluate the feasibility of the proposed orbital controller, in this section we present experimental results. Our control methodology uses the dynamic coupling between the passive joint and the active joints, and controls the active ones in order to bring the passive joint angle to a target orbit. Then, the orbit generator has been designed to generate balancing on Rabbit and not rely on specific joint trajectories. Our goal is to carry the Rabbit robot to a state of oscillation by means of our control law, and to reach a closed curve that produces a stable oscillating behaviour. A simple way to look at bipedal motion during slow walking, assuming only one foot on the ground, is a model employing the physics of an inverted pendulum, see Fig In this figure, the robot is represented by a point of mass equal to the total mass of the robot, positioned at the center of gravity and connected by a massless beam to the foot on the ground 8. 8 In this case, the ZMP is the point of contact with the ground.

91 74 CHAPTER 7. EXPERIMENTAL RESULTS ON RABBIT ROBOT Figure 7.6: The inverse pendulum equivalent system. The objective of this experiment is to show that our control law is able to stabilize the electromechanical system 9 at the upright position. Every time the center of mass of the robot passes the point in the middle of the feet towards a certain direction, then the appropriate leg is activated to resist the motion. If the center of gravity of the robot is not projected vertically onto the area of either foot, this requires a period of controlled instability in the balancing, which is difficult to accomplish. During the experimentation phase was noted that the performance of the controller is sensitive to incorrect initial conditions, parametric uncertainties, and friction forces. Due to the gravitational drift, the region of the state space where the robot can be kept in equilibrium is a reduced manifold. All these problems caused the robot s fall after some few seconds of successful orbital stability. 9 Rabbit robot represented by a multi-link inverted pendulum.

92 7.6. CONCLUSIONS Conclusions In this chapter, the control methodology presented in the last chapters was applied on a planar robot nicknamed Rabbit. When the system is underactuated, we cannot hope to order all degrees of freedom previously on definite trajectories. The existence of a non-actuated degree of freedom on a support-leg/ground contact has an important role in overall system stability. In balancing of bipedal robots always is possible to find a force whose influence causes the robot s fall down, then we cannot hope to achieve a globally stable balancing controller. It is necessary to consider a robust controller; robust in the sense that it can handle parametric uncertainties and friction forces, to avoid the robot s fall down.

93 76 CHAPTER 7. EXPERIMENTAL RESULTS ON RABBIT ROBOT

94 Chapter 8 General Conclusions and Future Research 8.1 General Conclusions Conclusions Underactuated electromechanical systems have proven to be very useful in a wide range of applications such as space and undersea robots, flexible and mobile robots, and many other current or future developments, because of their lesser weight and power consumption due to their lower number of actuators. Nevertheless, those systems have a noticeable disadvantage: they need special control techniques since usual control laws cannot handle the related dynamic and mathematical complexities. Even more, fully actuated mechanical systems can become into underactuated ones if an actuator fails, so it is far convenient to have a control for the underactuated system. We have addressed the problem of orbital stabilization of underactuated electromechanical systems, when the number of degrees of freedom is equal to the number of actuators minus one, establishing conditions for local convergence via an implicit algorithm, devised specially for this application, resulting in an entirely new control method. The proposed approach seeks to reach an isolated periodic orbit that produces stable oscillating behaviour on the full high-dimensional inverted pendulum, making use of the dynamic interaction between the passive joint and active joints. However, admissible limit cycles are not completely free because they should comply with the constraints imposed by the ground surface. The key characteristic of this strategy compared to many other closed-loop strategies proposed in the literature is that no reference information, such as desired trajectory or path, is required. 77

95 78 CHAPTER 8. GENERAL CONCLUSIONS AND FUTURE RESEARCH The mathematical model initially used by the LAPIN simulator to describe the dynamical behaviour of the Rabbit robot [Ro98] had to be corrected. The new mathematical model, introduced in this work and described in Appendix E, was used in the LAPIN simulator in order to model the new control law, and for the experiments carried on the Rabbit robot. In order to illustrate this method, we have chosen several underactuated systems: the Reaction Wheel Pendulum, the Acrobat Robot, the Gymnastic Robot, and the Rabbit Robot. We presented the results of several simulations which demonstrate the feasibility of the proposed control method, however, in real implementations it is necessary to consider a controller that can handle parametric uncertainties and friction forces, to avoid the robot s fall down. The approach followed allows to reach many links configurations. The work presented here can benefit practical problems such as the study of stable locomotion of human upper body for persons using leg prothesis, and bipedal robots. In the presented simulations, we have chosen initial conditions outside the limit cycle generated by the desired elliptic orbits O d ; however, when taking initial conditions inside the desired elliptic orbits, similar convergency results were observed. We worked with the zero-dynamics because we needed a simpler way to determine the stability of the nonlinear systems internal dynamics. However we must to be aware of the fact that even if local asymptotic stability of zero-dynamics is enough to guarantee the local asymptotic stability of the internal dynamics, and no results on the global stability or even large range stability can be drawn for internal dynamics of the studied systems, local stability is guaranteed for the internal dynamics only if the zero-dynamics is globally exponentially stable. 8.2 Future Research Further research on this kind of systems includes more accurate sizing of system parameters and further simulations utilizing these parameters, since the high complexity of such devices leads necessarily to a high degree of uncertainty (unmodelled dynamics) that has to be matched in order to get experimental results more similar to modelled behaviour. Future studies can also be performed to determine and to counteract effects of disturbances on the system, like friction, flex dynamics caused by cabling, torsion of the gears and the counter-balance bar, etc. Certain complex systems such as three-dimensional bipeds do not necessarily require complicated controls systems in order to accomplish the desired task, so the control methodology given in this thesis set the basis for the analysis of more complex kinds of

96 8.2. FUTURE RESEARCH 79 systems, as shown by the obtained results. Bipedal robots will be probably used for locomotion in rough terrain, terrestrial exploration, mining, forestry, industrial automation, operation in hazardous environments and farming, as well as prothesis development and testing, etc. However, the design and construction of feasible walking and running machines remains a challenge, and a better understanding of active balance is fairly required.

97 80 CHAPTER 8. GENERAL CONCLUSIONS AND FUTURE RESEARCH

98 Appendix C Background on Stability This appendix deals whit some definitions and background theory that will be used throughout this work. Stability analysis of dynamic systems is extremely relevant to characterize how a system reacts to small perturbations from a given equilibrium state. There are many kinds of stability concepts such as input-output stability, absolute stability, Lyapunov stability, and stability of periodic solutions [Ka96]. To characterize and study stability of an equilibrium point, in section C.1 we give several standard definitions on stability. As until now Lyapunov stability theory is the fundamental tool for stability analysis of dynamic systems, in section C.2 we review some of the key concepts and results of Lyapunov stability theory for ordinary differential equations. C.1 Stability Definitions Consider a dynamical system which satisfies: ẋ = f(x, t), x(t 0 ) = x 0, x IR n, t 0 (C.1) and assuming that f(x, t) satisfies the standard conditions for the existence and uniqueness of solutions 1, this system (C.1) is said to be autonomous (time-invariant) if f does not depend on t, and non autonomous (time-variant), otherwise. C.1.1 Equilibrium point Definition. A point x is called an equilibrium point of (C.1) if f(x, t) 0. For convenience, but without loss of generality, we may assume that the equilibrium point of interest occurs at x = 0. Therefore, by shifting the origin of the system (via a change of variable), we shall always study stability in the origin x = 0. If multiple equilibrium points exist, we will need to study the stability of each one by appropriately shifting the origin. t. 1 i.e., f(x, t) is Lypschitz continuous with respect to x, uniformly in t, and piecewise continuous in 147

99 148 APPENDIX C. BACKGROUND ON STABILITY C.1.2 Stability Definition. x = 0 is a stable equilibrium point of (C.1), if the trajectory x(t) remains close to 0 if the initial condition x 0 is close to 0. C.1.3 Stability in the Sense of Lyapunov Definition. The equilibrium point x = 0 of (C.1) is called a stable (in the sense of Lyapunov), if for all t 0 0, and ɛ > 0, there exists δ(t 0, ɛ) such that x 0 < δ(t 0, ɛ) x(t) < ɛ, t t 0 (C.2) where x(t) is the solution of (C.1) starting from x 0 at t 0. C.1.4 Uniform Stability Definition. x = 0 is called an uniformly stable equilibrium point of (C.1), if for all t 0 0, and ɛ > 0, there exists δ(t 0, ɛ) such that (C.2) for all t t 0, where x(t) is the solution of (C.1) starting from x 0 at t 0, and if δ can be chosen independently of t 0. C.1.5 Asymptotic Stability Definition. x = 0 is called an asymptotically stable equilibrium point of (C.1), if: x = 0 is a stable equilibrium point of (C.1). x = 0 is attractive, that is, for all t 0 0, there exists δ(t 0 ), such as that: x 0 < δ lim t x(t) = 0 (C.3) C.1.6 Uniform Asymptotic Stability Definition. x = 0 is called an uniform asymptotically stable (u.a.s.) equilibrium point of (C.1), if: x = 0 is a uniformly stable equilibrium point of (C.1).

100 C.2. LYAPUNOV STABILITY THEORY 149 the trajectory x(t) converges to 0 uniformly in t 0. More precisely, there exits δ > 0 and a function γ(τ, x 0 ) : IR + IR n IR +, such that lim τ γ(τ, x 0 ) = 0 for all x 0 and x 0 < δ x(t) γ(t t 0, x 0 ), t t 0 0. (C.4) C.1.7 Global Asymptotic Stability Definition. x = 0 is called a globally asymptotically stable equilibrium point of (C.1), if it is asymptotically stable and lim t x(t) = 0, x 0 IR n. C.1.8 Exponential Stability, Rate of Convergence Definition. x = 0 is called an exponentially stable equilibrium point of (C.1) if there exist m, α > 0 such that the solution x(t) satisfies: x(t) me α(t t 0) x 0, x 0 B h, t t 0 0 (C.5) B h is the closed ball of radius h centered at 0 in IR n, and the constant α is called the rate of convergence. C.2 Lyapunov Stability Theory In the study of Dynamical Systems there are different kinds of stability problems. Lyapunov stability theorem is a powerful tool which gives sufficient conditions for stability, asymptotic stability, uniform asymptotic stability, global asymptotic stability, etc. However, these theorems do not say whether the given conditions are also necessary. Lyapunov s direct method allows us to determine the nature of stability of an equilibrium point of (C.1) without explicitly integrating the differential equation. This method is basically a generalization of the idea that if there is some measure of the energy in a system, then we can study the rate of change of the energy of the system to ascertain stability. If this energy is decreasing, then the system will tend to its equilibrium. Definition. Class K Functions: A function α(ɛ) : IR + IR + belongs to the class K (denoted α(.) K), if it is continuous, strictly increasing, and α(0) = 0. Definition. Class KR Functions: A function α(ɛ) : IR + IR + belongs to the class KR if α(.) K and α(p) as p.

101 150 APPENDIX C. BACKGROUND ON STABILITY Definition. Locally Positive Definite Functions: A continuous function V (x, t) : IR + IR n IR + is called a locally positive definite function (l.p.d.f.) if, for some h > 0, and some α(.) K. V (t, 0) and V (x, t) α( x ), x B h, t 0. Definition. Positive Definite Functions: A continuous function V (x, t) : IR + IR n IR + is called a positive definite function (p.d.f.) if, for some α(.) K V (t, 0) and V (x, t) α( x ), x IR n, t 0 and the function α(p) as p. Definition. Decrescent Function: The function V (x, t) is called decrescent, if there exists a function β(.) K, such that: V (x, t) β( x ), x B h, t 0. Theorem. Basic Theorem of Lyapunov: Let V (x, t) be a non-negative function 2 with derivative V (x, t) along the trajectories of the system (C.1). 1. If V (x, t) is locally positive definite and V (x, t) 0 locally in x and for all t, then the origin of the system is locally stable (in the sense of Lyapunov). 2. If V (x, t) is locally positive definite and decrescent, and V (x, t) 0 locally in x and for all t, then the origin of the system is uniformly locally stable (in the sense of Lyapunov). 3. If V (x, t) is locally positive definite and decrescent, and V (x, t) is locally positive definite, then the origin of the system is uniformly locally asymptotically stable. 4. If V (x, t) is positive definite and decrescent, and V (x, t) is positive definite, then the origin of the system is globally uniformly asymptotically stable. Conditions on Conditions on Conclusions V (x, t) V (x, t) l.p.d.f. 0 locally Stable l.p.d.f., decrescent 0 locally U nif ormly stable l.p.d.f. l.p.d.f. Asymptotically stable l.p.d.f., decrescent l.p.d.f. U nif ormly asymptotically stable p.d.f., decrescent p.d.f. Globally u.a.s. Table C.1: Summary of the basic theorem of Lyapunov. Now, consider the following autonomous system: ẋ = f(x), x(0) = x 0 (C.6) 2 i.e., V (x, t) is said to be negative definite if V (x, t) is positive definite.

102 C.2. LYAPUNOV STABILITY THEORY 151 Definition. Invariant Set: A set M is called an invariant set with respect to the dynamics (C.6) if x(0) M x(t) M t IR. Definition. Positively Invariant Set: A set M is called positively invariant if: x(0) M x(t) M t 0. Theorem. Continuous Dependence on Initial Conditions and Parameters: Suppose that f(x, λ, t) is locally Lipschitz in x, uniformly λ as well as t, over a domain D Λ [t 0, T ] where D is an open connected subset of IR n and Λ = {λ λ - λ 0 c}, for some c > 0. Let x 1 (λ 0, t) be a nominal solution of: ẋ = f(x, λ 0, t), x(t 0 ) = x 0 (C.7) that is, the solution for some nominal initial condition x(t 0 ) and set of parameter values λ. 3 Then, given ɛ > 0, there exist δ x > 0 and δ λ > 0 such that for any off-nominal x initial condition and off-nominal set of parameter values λ satisfying: the unique solution x 2 ( λ, t) of: x 0 - x < δ x, and λ 0 - λ < δ λ ẋ = f(x, λ, t), x(t 0 ) = x satisfies x 1 (λ 0, t) - x 2 ( λ, t) < ɛ for all t [t 0, T ]. Theorem. LaSalle s Invariance Principle: Let Ω D be a compact set (i.e., closed and bounded) 4, that is positively invariant with respect to the dynamics (C.6). Let V(.) be a continuously differentiable function on D such that V (x) 0 in Ω. Let E be the set of all points in Ω where V (x) = 0 and let M be the largest invariant set contained in E. Then every solution starting in Ω converges to M as t. 3 Assume that x1(λ 0, t) remains within D for all t [t 0, T ]; otherwise, choose T smaller. 4 A set S IR n is open if every point in S is contained in an open ball of points which are also in S. A set is closed if its complement is open. A set is bounded if the entire set can be contained in a closed ball of nonzero radius.

103 152 APPENDIX C. BACKGROUND ON STABILITY

104 Appendix D Equations of Motion for Three-DOF Model of Gymnastic Robot The presented example in Chapter 5 consider a Three-Link Planar Robot, depicted in Fig. 5.1, which is free to swing in the vertical plane. Below, we give the explicit form of the various terms appearing in the dynamic equation of this Three-Link Planar Mechanism used in our study. D.1 Generic elements of the inertia matrix M(q) m 11 = k 1 + k 2 + k (k 3 + k 7 ) cos(q a1 ) + 2 k 5 cos(q a1 + q a2 ) + 2 k 6 cos(q a2 ) m 12 = k 2 + k 8 + (k 3 + k 7 ) cos(q a1 ) + k 5 cos(q a1 + q a2 ) + 2 k 6 cos(q a2 ) m 13 = k 9 + k 5 cos(q a1 + q a2 ) + k 6 cos(q a2 ) m 21 = m 12 m 22 = k 2 + k k 6 cos(q a2 ) m 23 = k 9 + k 6 cos(q a2 ) m 31 = m 13 m 32 = m 23 m 33 = k 9 153

105 154 APPENDIX D. EQUATIONS OF MOTION FOR THREE-DOF MODEL OF GYMNASTIC ROBOT where: k 1 = m 1 l 2 c1 + m 2 l I 1 k 2 = m 2 l 2 c2 + I 2 k 3 = m 2 l 1 + l c2 k 4 = m 3 (l 2 c3 + l l 2 2) + I 3 k 5 = m 3 l 1 l c3 k 6 = m 3 l 2 l c3 k 7 = m 3 l 1 l 2 k 8 = m 3 (l 2 c3 + l 2 2) + I 3 k 9 = m 3 l 2 c3 + I 3 D.2 Generic elements of the Coriolis and centrifugal torques c 11 = f 5 q a1 + f 6 q a2 c 12 = f 5 ( q p + q a1 ) + f 6 q a2 c 13 = f 6 ( q p + q a1 + q a2 ) c 21 = f 5 q a1 + f 4 q a2 c 22 = f 4 q a2 c 23 = f 4 ( q p + q a1 + q a2 ) c 31 = f 5 q a1 f 4 q a2 c 32 = f 4 ( q p + q a1 ) c 33 = 0 g 1 = g ((m 11 l c1 + 2 m 12 l 1 ) cos(q p ) + (m 12 l c2 + m 2 l 2 ) cos(q p + q a1 ) + m 2 l c3 cos(q p + q a1 + q a2 )) g 2 = g ((m 12 l c2 + m 2 l 2 ) cos(q p + q a1 ) + m 2 l c3 cos(q p + q a1 + q a2 )) g 3 = g m 2 l c3 cos(q p + q a1 + q a2 )) where: f 1 = k 3 sin(q a1 ) f 2 = k 7 sin(q a1 ) f 3 = k 5 sin(q a1 + q a2 ) f 4 = k 6 sin(q a2 ) f 5 = f 1 + f 2 + f 3 f 6 = f 3 + f 4 g = 9.8 [kg m/s 2 ]

106 D.3. GENERIC ELEMENTS OF THE EQUATION WHICH REPRESENTS THE ZERO-DYNAMICS 155 D.3 Generic elements of the equation which represents the zero-dynamics For: β 2 (q p, φ, θ) q p + β 0 (q p, q p, φ, θ, θ, θ) = 0 (D.1) whose solution will be well defined if β 2 (q p, θ) > 0, we have: β 2 (q p, φ, θ) = m 11 + m 12 a 1 + m 13 a 2 m 11 (q p, φ, θ) = k 1 + k 2 + k (k 3 + k 7 ) cos(a 1 q p + b 1 ) + 2 k 5 cos((a 1 + a 2 ) q p + b 1 + b 2 ) + 2 k 6 cos(a 2 q p + b 2 ) m 12 (q p, φ, θ) = k 2 + k 8 + (k 3 + k 7 ) cos(a 1 q p + b 1 ) + k 5 cos((a 1 + a 2 ) q p + b 1 + b 2 ) + 2 k 6 cos(a 2 q p + b 2 ) m 13 (q p, φ, θ) = k 9 + k 5 cos((a 1 + a 2 ) q p + b 1 + b 2 ) + k 6 cos(a 2 q p + b 2 ) β 0 (q p, q p, φ, θ, θ, θ) = (f 5 (1 + 2 a 1 ) + f 6 a 2 (a (1 + a 1 ))) q 2 p + 2 (f 5 ḃ1 (1 + a 1 ) + f 6 a 2 ḃ1) q 1 + f 5 ḃ2 1 + m 12 b 1 + g ((m 11 l c1 + 2 m 12 l 1 ) cos(q p ) + (m 12 l c2 + m 2 l 2 ) cos((1 + a 1 ) q p + b 1 ) + m 2 l c3 cos( (1 + a 1 + a 2 ) q p + b 1 + b 2 ) f 1 (q p, φ, θ) = k 3 sin(a 1 q p + b 1 ) f 2 (q p, φ, θ) = k 7 sin(a 1 q p + b 1 ) f 3 (q p, φ, θ) = k 5 sin((a 1 + a 2 ) q p + b 1 + b 2 ) f 4 (q p, φ, θ) = k 6 sin(a 2 q p + b 2 ) f 5 (q p, φ, θ) = f 1 (q p, φ, θ) + f 2 (q p, φ, θ) + f 3 (q p, φ, θ) f 6 (q p, φ, θ) = f 3 (q p, φ, θ) + f 4 (q p, φ, θ) The zero-dynamics (5.11) is forced to behaves as the orbit generator (2.41) if we can find an adaptation law for θ, such that the following differential equation is satisfied: β(x, φ, θ, θ, θ) = β d (x), t 0 (D.2) Let θ 1 = b 1, θ 2 = θ 1, then the adaptation law takes, in our particular case, the form: θ 1 = θ 2 1 θ 2 = (f m 12 (x 1,θ) 5(x 1, θ) θ2 2 + f 7 (x 1, θ) x 2 θ 2 + f 8 (x 1, θ) x g 1 (x 1, θ) + β 2 (x 1, θ) β d (x)) (D.3)

107 156 APPENDIX D. EQUATIONS OF MOTION FOR THREE-DOF MODEL OF GYMNASTIC ROBOT where: θ 2 (q p, φ, θ, θ, β d ) = 1 m 12 (x 1, θ) f(q p, φ, θ, θ, β d ) f(q p, φ, θ, θ, β d ) = β 0 β d + (f 5 (1 + 2 a 1 ) + f 6 a 2 (a (1 + a 1 ))) q 2 p + 2 (f 5 ḃ1 (1 + a 1 ) + f 6 a 2 ḃ1) q 1 + f 5 ḃ2 1 + g ((m 11 l c1 + 2 m 12 l 1 ) cos(q p ) + (m 12 l c2 + m 2 l 2 ) cos((1 + a 1 ) q p + b 1 ) + m 2 l c3 cos((1 + a 1 + a 2 ) q p + b 1 + b 2 ) f 1 (x 1, θ) = k 3 sin(a 1 (x 1 + q d p) + b 1 ) f 2 (x 1, θ) = k 7 sin(a 1 (x 1 + q d p) + b 1 ) f 3 (x 1, θ) = k 5 sin((a 1 + a 2 ) (x 1 + q d p) + b 1 + b 2 ) f 4 (x 1, θ) = k 6 sin(a 2 (x 1 + q d p) + b 2 ) f 5 (x 1, θ) = f 1 (x 1, θ) + f 2 (x 1, θ) + f 3 (x 1, θ) f 6 (x 1, θ) = f 3 (x 1, θ) + f 4 (x 1, θ) f 7 (x 1, θ) = 2 ((1 + a 1 ) f 5 (x 1, θ) + a 2 f 6 (x 1, θ)) f 8 (x 1, θ) = (1 + 2 a 1 ) f 5 (x 1, θ) + a 2 (a (1 + a 1 ) f 6 (x 1, θ)) g 1 (x 1, θ) = g ((m 11 l c1 + 2 m 12 l 1 ) cos(x 1 + q d p) + (m 12 l c2 + m 2 l 2 ) cos((1 + a 1 ) (x 1 + q d p) + b 1 ) + m 2 l c3 cos((1 + a 1 + a 2 ) (x 1 + q d p) + b 1 + b 2 )) β 2 (x 1, θ) = m 11 (x 1, θ) + a 1 m 12 (x 1, θ) + a 2 m 13 (x 1, θ) m 11 (x 1, θ) = k 1 + k 2 + k (k 3 + k 7 ) cos(a 1 (x 1 + q d p) + b 1 ) + 2 k 5 cos((a 1 + a 2 ) (x 1 + q d p) + b 1 + b 2 ) + 2 k 6 cos(a 2 (x 1 + q d p) + b 2 ) m 12 (x 1, θ) = k 2 + k 8 + (k 3 + k 7 ) cos(a 1 (x 1 + q d p) + b 1 ) + k 5 cos((a 1 + a 2 ) (x 1 + q d p) + b 1 + b 2 ) + 2 k 6 cos(a 2 (x 1 + q d p) + b 2 ) m 13 (x 1, θ) = k 9 + k 5 cos((a 1 + a 2 ) (x 1 + q d p) + b 1 + b 2 ) + k 6 cos(a 2 (x 1 + q d p) + b 2 )

108 Appendix E Equations of Motion for Five-DOF Model of Rabbit E.1 Kinematic Model Rabbit, our planar biped walker, is a robot that locomotes on the sagittal plane with one degree of underactuation. Rabbit has seven DOF, two legs, two knees, and a trunk. As this biped is driven by four torques, only four independent variables can be chosen. Let define q, a set of angular coordinates describing the configuration of this robot, see Fig. E.1: q = q 1 q 2 q 3 q 4 q 5 q 6 q 7 = θ 1 θ 31 θ 41 θ 32 θ 42 x z (E.1) 157

109 158 APPENDIX E. EQUATIONS OF MOTION FOR FIVE-DOF MODEL OF RABBIT Z q 1 Z l c1 q 2 m 1 l 1 q 4 z z l31 m 31 l c31 l c32 m 32 l 32 q 3 l c41 l c42 q 5 l 41 m 41 l 42 m 42 Y Figure C1.1: Angular convention. x X Y Figure C1.2: Coordinate convention. x X Figure E.1: Rabbit Model. Nomenclature: l 1 : Length of trunk. l 3i : Length of femur i. l 4i : Length of tibia i. l c3i : Distance to center of mass of femur i along the center line. l c4i : Distance to center of mass of tibia i along the center line. m 1 : Mass of trunk. m 3i : Mass of femur i. m 4i : Mass of tibia i. I 1 : Moment of inertia of trunk about center of mass. I 3i : Moment of inertia of femur i about center of mass. : Moment of inertia of tibia i about center of mass. I 4i with: i = 1, 2.

110 E.2. DYNAMIC MODEL 159 E.2 Dynamic Model Using the method of Lagrange-Euler, the dynamic model is written in the form [SV89, Sh96, Ro98]: D(q(t)) q(t) + C(q(t), q(t)) q(t) + g(q(t)) = S u (E.2) with: D : Symmetric positive definite generalized inertia matrix R 7x7. C : Matrix of Coriolis and centrifugal terms R 7x7. g : Vector of gravitational terms R 7. q : Vector of positions R 7. q : Vector of speeds R 7. q : Vector of accelerations R 7. S : Matrix that defines the actions of the motors on the articulations R 7x4. u : Vector of generalized forces R 4. Let define a torque vector u, performing work on the active degrees of freedom. u = u 1 u 2 u 3 u 4 (E.3) Torques are applied between each connection of two link, but not between the support leg and the walking surface, then we can define the following matrix: S = (E.4)

111 160 APPENDIX E. EQUATIONS OF MOTION FOR FIVE-DOF MODEL OF RABBIT Z u 1 z u 2 u 3 u 4 Y x X Figure E.2: Rabbit Model. Torque convention. E.2.1 Impact Model The impact model in [HM94, GAP01] includes the impact between two rigid bodies, the swing leg and the walking surface. This model is used under the following hypotheses: A. The impact occurs in an infinitesimal time, i.e., instantaneous. B. The contact with the walking surface is perfectly inelastic, i.e., the impact between the swing leg and the walking surface results in no slipping and no rebound of the swing leg. C. The dynamic coupling between longitudinal and lateral plane is neglected. D. The transfer of support from the stance leg and swing leg is instantaneous. E. At time of impact the external forces are considered as impulses. F. During the impact the actuators do not generate impulses. G. The torques supplied by the actuators are not impulsional. H. The impulsive forces may result in an instantaneous change in the velocities, but there is no instantaneous change in the positions. Let us add the external forces acting on the robot at the contact point, then from Eq. (E.2) one obtains: D(q(t)) q(t) + C(q(t), q(t)) q(t) + g(q(t)) = S u + J T e (q(t)) F e (q(t), q(t)) (E.5)

112 E.2. DYNAMIC MODEL 161 where: J e : Jacobian Matrix R 4x7, constituted by the forces acting on the robot s joint. F e : Vector of external force R 4, constituted by the normal ground reaction forces (F eni ) and the tangential ground reaction forces (F eti ) applied on the femur i. F e = F en1 F et1 F en2 F et2 (E.6) From the last hypotheses, the angular momentum is conserved, then we integrate Eq. (E.5) on the impact duration, one can deduce the following equation: D(q(t)) ( q + q ) = I e (E.7) where I e is the vector of generalized forces produced by the impact, q + is the velocity just after impact and q is the velocity just before impact. By supposing that the impact between the swing leg and the walking surface results in no slipping and no rebound of the swing leg, this yields: x p = x l 3 senθ 3i l 4 sen(θ 3i + θ 4i ) = L z p = z + l 3 cos(θ 3i ) + l 4 cos(θ 3i + θ 4i ) = 0 (E.8) where L is the length of a step. Expressing the Eq. (E.8) in the form ϕ(q) = 0 (J e = ϕ q ), and evaluating the total time derivative of ϕ(q) we have: J e q + = 0 (E.9) Therefore, the dynamic model becomes: Single support period D(q(t)) q(t) + C(q(t), q(t)) q(t) + g(q(t)) = S u + J T e (q(t)) F e (q(t), q(t)) D(q(t)) ( q + q ) = I e (E.10) Double support period D(q(t)) q(t) + C(q(t), q(t)) q(t) + g(q(t)) = S u + J T e (q(t)) F e (q(t), q(t)) D(q(t)) ( q + q ) = 0 (E.11)

113 162 APPENDIX E. EQUATIONS OF MOTION FOR FIVE-DOF MODEL OF RABBIT F eni = { 0 if: zpi > 0 λ c ż pi k z pi if: z pi 0 (E.12) F eti = { 0 if: zpi > 0 λ c ẋ pi k (x pi x c ) if: z pi 0 (E.13) where k is stiffness constant, λ c is the spring constant, and x c it is the contact abscissa. Now, the result of solving Eq. (E.7) and Eq. (E.9) gives an expression for q + in terms of q, this is: [ ] [ ] q + q q + = q (E.14) where (.) is a function that relates the robot state value just after impact in terms of the state value jus before the impact. Therefore, we have a biped model robot which take into account the change in the velocities and the transfer of support from the stance leg and swing leg commuting the coordinates of legs.

114 E.3. KINEMATIC EQUATIONS 163 E.3 Kinematic Equations Defining: (x 1, z 1 ) : Position of center of mass of trunk. (x 3i, z 3i ) : Position of center of mass of femur i. (x 4i, z 4i ) : Position of center of mass of tibia i. (ẋ 1, ż 1 ) : Velocity of center of mass of trunk. (ẋ 3i, ż 3i ) : Velocity of center of mass of femur i. (ẋ 4i, ż 4i ) : Velocity of center of mass of tibia i. with: i = 1, 2. we have: E.3.1 Position Variables x 1 = x l 1 2 sen(θ 1 ) x 31 = x l 3 2 sen(θ 31 ) x 32 = x l 3 2 sen(θ 32 ) x 41 = x l 3 sen(θ 31 ) l 4 2 sen(θ 31 + θ 41 ) x 42 = x l 3 sen(θ 32 ) l 4 2 sen(θ 32 + θ 42 ), z 1 = z + l 1 2 cos(θ 1 ) z 31 = z + l 3 2 cos(θ 31 ) z 32 = z + l 3 2 cos(θ 32 ) z 41 = z + l 3 cos(θ 31 ) + l 4 2 cos(θ 31 + θ 41 ) z 42 = z + l 3 cos(θ 32 ) + l 4 2 cos(θ 32 + θ 42 ) (E.15) (E.16)

115 164 APPENDIX E. EQUATIONS OF MOTION FOR FIVE-DOF MODEL OF RABBIT E.3.2 Velocity Variables ẋ 1 = ẋ l 1 2 q 1 cos(θ 1 ) ẋ 31 = ẋ l 3 2 θ 31 cos(θ 31 ) ẋ 32 = ẋ l 3 2 θ 32 cos(θ 32 ) ẋ 41 = ẋ l 3 θ 31 cos(θ 31 ) l 4 2 ( θ 31 + θ 41 ) cos(θ 31 + θ 41 ) ẋ 42 = ẋ l 3 θ 32 cos(θ 32 ) l 4 2 ( θ 32 + θ 42 ) cos(θ 32 + θ 42 ), ż 1 = ż l 1 2 q 1 sen(θ 1 ) ż 31 = ż l 3 2 θ 31 sen(θ 31 ) ż 32 = ż l 3 2 θ 32 sen(θ 32 ) ż 41 = ż l 3 θ 31 sen(θ 31 ) l 4 2 ( θ 31 + θ 41 ) sen(θ 31 + θ 41 ) ż 42 = ż l 3 θ 32 sen(θ 32 ) l 4 2 ( θ 32 + θ 42 ) sen(θ 32 + θ 42 ) (E.17) (E.18) E.4 Dynamic Equations E.4.1 Kinetic Energy K = n k i = 1 2 i=1 n (m i vgi 2 + I i ωi 2 ) (E.19) i=1 where: k i : Kinetic energy on the link i. v gi : Lineal velocity of the gravity center on the link i. ω 2 i : Rotational velocity on the link i. with: i = 1, 2... n. where n represents the number of DOF s, for Rabbit i = 5, and using m i and I i defined in section E.1 (Kinematic Model). The robot kinetic energy can be given by [HW79, SV89, Sl91, Ro98]: K = 1 2 qt D(q) q (E.20) where: d ii represents the coefficients q 2 i in the expression q T D(q) q.

116 E.4. DYNAMIC EQUATIONS 165 The value of d ij represents half of the value of the coefficients q i q j in q T D(q) q, with i j, because q i q j = q j q i. From Eq. (E.20), and defining m 3 = m 31 = m 32, m 4 = m 41 = m 42 ; I 3 = I 31 = I 32, I 4 = I 41 = I 42, the robot kinetic energy gives: K = 1 2 [(m 1 ((ẋ l 1 2 θ 1 cos(θ 1 )) 2 +(ż l 1 2 θ 1 sen(θ 1 )) 2 )+m 3 ((ẋ l 3 2 θ 31 cos(θ 31 )) 2 + (ż l 3 2 θ 31 sen(θ 31 )) 2 ) + m 3 ((ẋ l 3 2 θ 32 cos(θ 32 )) 2 + (ż l 32 θ 32 sen(θ 32 )) 2 )+m 4 ((ẋ l 3 θ 31 cos(θ 31 ) l 4 2 ( θ 31 + θ 41 ) cos(θ 31 + (E.21) θ 41 )) 2 +(ż l 3 θ 31 sen(θ 31 ) l 4 2 ( θ 31 + θ 41 ) sen(θ 31 +θ 41 )) 2 )+m 4 ((ẋ l 3 θ 32 cos(θ 32 ) l 4 2 ( θ 32 + θ 42 ) cos(θ 32 + θ 42 )) 2 + (ż l 3 θ 32 sen(θ 32 ) l 42 ( θ 32 + θ 42 ) sen(θ 32 +θ 42 )) 2 )+I 1 θ 1 2 +(( θ 31 +I 3 ( θ θ 32)+I 2 4 θ 41 ) 2 + ( θ 32 + θ 42 ) 2 )] Finally, from Eq. (E.21) we have: K = 1 2 [(m 1 l I 1 ) θ 2 1 +(m 3 l m 4 (l l2 4 4 )+m 4 l 3 l 4 cos(θ 31 ) cos(θ 31 + (E.22) θ 41 )+m 4 l 3 l 4 sen(θ 31 ) sen(θ 31 +θ 41 )+I 3 +I 4 ) θ (m 3 l m 4 (l3 2 + l4 2 4 )+m 4 l 3 l 4 cos(θ 32 ) cos(θ 32 +θ 42 )+m 4 l 3 l 4 sen(θ 32 ) sen(θ 32 +θ 42 )+ I 3 +I 4 ) θ (m 4 l I 4 ) θ (m 4 l I 4 ) θ (m 1 +2 m 3 +2 m 4 ) ẋ 2 +(m 1 +2 m 3 +2 m 4 ) ż 2 (m 1 l 1 cos(θ 1 ) θ 1 ẋ (m 1 l 1 sen(θ 1 )) θ 1 ż+ (2 m 4 l m 4 l 3 l 4 cos(θ 31 ) cos(θ 31 +θ 41 )+m 4 l 3 l 4 sen(θ 31 ) sen(θ 31 + θ 41 ) + 2 I 4 ) θ 31 θ 41 (m 3 l 3 cos(θ 31 ) + m 4 2 l 3 cos(θ 31 ) + m 4 l 4 cos(θ 31 + θ 41 )) θ 31 ẋ (m 3 l 3 sen(θ 31 ) + m 4 2 l 3 sen(θ 31 ) + m 4 l 4 sen(θ 31 + θ 41 )) θ 31 ż m 4 l 4 cos(θ 31 + θ 41 ) θ 41 ẋ m 4 l 4 sen (θ 31 +θ 41 ) θ 41 ż+(2 m 4 l m 4 l 3 l 4 cos(θ 32 ) cos(θ 32 +θ 42 )+m 4 l 3 l 4 sen(θ 32 ) sen(θ 32 + θ 42 ) + 2 I 4 ) θ 32 θ 42 m 4 l 4 cos(θ 32 + θ 42 ) θ 42 ẋ m 4 l 4 sen(θ 32 +θ 42 ) θ 42 ż (m 3 l 3 cos(θ 32 )+m 4 2 l 3 cos(θ 32 )+ m 4 l 4 cos(θ 32 + θ 42 )) θ 32 ẋ (m 3 l 3 sen(θ 32 ) + m 4 2 l 3 sen(θ 32 ) + m 4 l 4 sen(θ 32 + θ 42 )) θ 32 ż] Therefore, from Eq. (E.20) and Eq. (E.22) the inertial matrix is:

117 166 APPENDIX E. EQUATIONS OF MOTION FOR FIVE-DOF MODEL OF RABBIT D(q) = d d 16 d 17 0 d 22 d d 26 d 27 0 d 32 d d 36 d d 44 d 45 d 46 d d 54 d 55 d 56 d 57 d 61 d 62 d 63 d 64 d 65 d 66 0 d 71 d 72 d 73 d 74 d 75 0 d 77 (E.23) where: d 11 = m 1 l I 1 d 16 = m 1 l1 2 cos(θ 1 ) d 17 = m 1 l1 2 sen(θ 1 ) d 22 = m 3 l m 4 (l l l 3 l 4 cos(θ 41 )) + I 3 + I 4 d 23 = m 4 ( l l 3 l 4 2 cos(θ 41 )) + I 4 d 26 = m 3 l3 2 cos(θ 31 ) + m 4 (l 3 cos(θ 31 ) + l 4 2 cos(θ 31 + θ 41 )) d 27 = m 3 l3 2 sen(θ 31 ) + m 4 (l 3 sen(θ 31 ) + l 4 2 sen(θ 31 + θ 41 )) d 32 = d 23 d 33 = m 4 l I 4 d 36 = m 4 l4 2 cos(θ 31 + θ 41 ) d 37 = m 4 l4 2 sen(θ 31 + θ 41 ) d 44 = m 3 l m 4 (l l l 3 l 4 cos(θ 42 )) + I 3 + I 4 d 45 = m 4 ( l l 3 l 4 2 cos(θ 42 )) + I 4 d 46 = m 3 l3 2 cos(θ 32 ) + m 4 (l 3 cos(θ 32 ) + l 4 2 cos(θ 32 + θ 42 )) d 47 = m 3 l3 2 sen(θ 32 ) + m 4 (l 3 sen(θ 32 ) + l 4 2 sen(θ 32 + θ 42 )) d 54 = d 45 d 55 = m 4 l I 4 d 56 = m 4 l4 2 cos(θ 32 + θ 42 ) d 57 = m 4 l4 2 sen(θ 32 + θ 42 ) d 61 = d 16 d 62 = d 26 d 63 = d 36 d 64 = d 46 d 65 = d 56

118 E.4. DYNAMIC EQUATIONS 167 d 66 = m m m 4 d 71 = d 17 d 72 = d 27 d 73 = d 37 d 74 = d 47 d 75 = d 57 d 77 = m m m 4 E.4.2 Potential Energy The robot s potential energy can be expressed as [SV89], [Sh96], and [Ro98]: P = n p i = g i=1 n i=1 m i z gi (E.24) where: g : Vector of gravity. z gi : Distance at the gravity center of link i. with: i = 1, From (E.24), the Rabbit s potential energy is: P = g [(m m m 4 ) z + m 1 l1 2 cos(θ 1 ) + m 3 l3 2 (cos(θ 31 ) + cos(θ 32 )) + As g(q) = P q m 4 l 3 (cos(θ 31 ) + cos(θ 32 )) + m 4 l4 2 (cos(θ 31 + θ 41 ) + cos(θ 32 + θ 42 ))] [SV89], we have: (E.25) g(q) = m 1 l1 2 sen(θ 1 ) m 3 l3 2 sen(θ 31 ) m 4 l 3 sen(θ 31 ) m 4 l4 2 sen(θ 31 + θ 41 ) m 4 l4 2 sen(θ 31 + θ 41 ) m 3 l3 2 sen(θ 32 ) m 4 l 3 sen(θ 32 ) m 4 l4 2 sen(θ 32 + θ 42 ) m 4 l4 2 sen(θ 32 + θ 42 ) 0 (E.26) m m m 4

119 168 APPENDIX E. EQUATIONS OF MOTION FOR FIVE-DOF MODEL OF RABBIT The Coriolis and centrifugal terms for this robot are: C(q, q) = c 22 c c c 44 c c c 61 c 62 c 63 c 64 c c 71 c 72 c 73 c 74 c (E.27) where: c 22 = m 4 l3 l 4 2 sen(θ 41 ) θ 41 c 23 = m 4 l3 l 4 2 sen(θ 41 ) ( θ 31 + θ 41 ) c 32 = m 4 l3 l 4 2 sen(θ 41 ) θ 31 c 44 = m 4 l3 l 4 2 sen(θ 42 ) θ 42 c 45 = m 4 l3 l 4 2 sen(θ 42 ) θ 42 c 54 = m 4 l3 l 4 2 sen(θ 42 ) θ 32 c 61 = m 1 l1 2 sen(θ 1 ) θ 1 c 62 = (m 3 l3 2 sen(θ 31 ) + m 4 (l 3 sen(θ 31 ) + l 4 2 sen(θ 31 + θ 41 ))) θ 31 c 63 = m 4 l4 2 sen(θ 31 + θ 41 ) ( θ 31 + θ 41 ) c 64 = (m 3 l3 2 sen(θ 32 ) + m 4 (l 3 sen(θ 32 ) + l 4 2 sen(θ 32 + θ 42 ))) θ 32 c 65 = m 4 l4 2 sen(θ 32 + θ 42 ) θ 42 c 71 = m 1 l1 2 cos θ 1 θ 1 c 72 = (m 3 l3 2 cos(θ 31 ) + m 4 (l 3 cos(θ 31 ) + l 4 2 cos(θ 31 + θ 41 ))) θ 31 + m 4 l4 2 cos(θ 31 + θ 41 ) θ 41 c 73 = m 4 l4 2 cos (θ 31 + θ 41 ) ( θ 31 + θ 41 ) c 74 = (m 3 l3 2 cos(θ 32 ) + m 4 (l 3 cos(θ 32 ) + l 4 2 cos(θ 32 + θ 42 ))) θ 32 + m 4 l4 2 cos(θ 32 + θ 42 ) θ 42 c 75 = m 4 l4 2 cos(θ 32 + θ 42 ) ( θ 32 + θ 42 )

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122 BIBLIOGRAPHY 171 [Mc90] T. McGeer, Passive Dynamic Walking, International Journal of Robotics Research, vol. 9, pp , [MFG03] [MS84] [Ol01] [PC00] [PDP97] F.J. Muñoz-Almaraz, E. Freire, and J. Galán, Bifurcations and Control of Periodic Orbits in Symmetric Hamiltonian Systems: An Application to the Furuta Pendulum, 2 nd Workshop Lagrangian and Hamiltonian Methods for Nonlinear Control, Seville, Spain, pp , H. Miura, and I. Shimoyama, Dynamic Walk of a Biped, International Journal of Robotics Research, vol. 3, pp , R. Olfati-Saber, Global Stabilization of a Flat Underactuated System: the Inertia Wheel Pendulum, 40 th Proceedings of the IEEE Conference on Decision and Control, Orlando, Florida, USA, J.H. Park, and H.C. Cho, An On-Line Trajectory Modifier for the Base Link of Biped Robots to Enhance Locomotion Stability, IEEE International Conference on Robotics and Automation, ICRA, pp , J. Pratt, P. Dilworth, and G. Pratt, Virtual Model Control of a Bipedal Walking Robot, IEEE International Conference on Robotics and Automation, ICRA, pp , [PK98] J.H. Park, and K.D. Kim, Biped Robot Walking Using Gravity- Compensated Inverted Pendulum Mode and Computed Torque Control, IEEE International Conference on Robotics and Automation, ICRA, pp , [PSCG03] J.W. Perram, A. Shiriaev, C. Canudas-de-Wit, and F. Grognard, Explicit Formula for a General Integral of Motion for a Class of Mechanical Systems Subject to Holonomic Constraint, 2 nd Workshop Lagrangian and Hamiltonian Methods for Nonlinear Control, Seville, Spain, pp , [Ro94] M.E. Rosheim, Robot Evolution: The Development of Anthrobotics, John Wiley and Sons, pp. 255, [Ro98] [SCL01] [Sh96] L. Roussel, Génération de Trajectories Optimales de Marche pour un Robot Bipède, Thèse de Doctorat, Institut National Politechnique de Grenoble, France, M.W. Spong, P. Corke, and R. Lozano, Nonlinear Control of the Reaction Wheel Pendulum, Automatica, vol. 37(11), pp , C. Shih, The Dynamics and Control of a Biped Walking Robot with Seven Degrees of Freedom, Trans. ASME J. Dyn. Syst. Measurement Control, Vol. 118, pp , [Sl91] J. Slotine, Applied Nonlinear Control, Prentice-Hall, New Jersey, [SV89] M.W. Spong, and M. Vidyasagar, Robot Dynamics and Control, John Wiley and Sons, New York, 1989.

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124 Appendix A Résumé A.1 Stabilisation Orbitale de Systèmes Sous- Actionnés Nous présentons maintenant notre recherche sur la stabilisation orbitale des systèmes électromécaniques à n-degrés de liberté (DDL) avec 1 degré de sous-actionnement. La plupart des robots bipèdes actuellement élaborés assurent, au moyen de moteurs localisés aux chevilles, le moment de rotation nécessaire à leur démarche. Cependant, à cause de leur structure, il est nécessaire que ces robots aient de grands pieds. Pour éviter cet inconvénient, qui rend hasardeuse la marche sur des terrains irréguliers, une solution est d équilibrer le système en utilisant seulement des actionneurs aux genoux et en laissant les chevilles non actionnées. Cela rendrait possible la marche d un robot bipède sur divers types de terrains (même accidentés), grâce au support ponctuel de la jambe sur l espace de travail. Voilà précisément ce qui a motivé notre étude du problème de la stabilisation orbitale des systèmes sous-actionnés. Dans les robots sous-actionnés, du fait de la présence d articulations passives, la dynamique est beaucoup plus complexe que dans les systèmes complètement actionnés. Cette présence d articulations passives se justifie principalement par trois raisons: Nous pouvons ainsi commander un robot malgré le défaut des actionneurs. Nous pouvons utiliser moins d énergie pour le faire travailler. Rendre intentionnellement passives quelques articulations est un gage de sécurité vis-à-vis des personnes alentours. Ces systèmes électromécaniques non holonomiques se présentent sous plusieurs formes à applications multiples, comme les robots sous-marins, les robots utilisés dans l espace, les robots flexibles, les robots mobiles, et d autres machines qui imitent la locomotion animale, comme les robots marcheurs. Généralement, les systèmes électromécaniques sous-actionnés ont des points d équilibre qui dépendent de leurs paramètres cinétiques et dynamiques [DMO00]. 81

125 82 APPENDIX A. RÉSUMÉ La marche est l une des actions de l être humain les plus communes ; elle exige de l humain un contrôle complexe pour être efficace, même si elle nécessite peu d energie [Mc90]. La locomotion par les jambes a été employée comme un mécanisme de locomotion biologique durant des millions d années. Depuis quelque temps, les chercheurs ont combiné des observations scientifiques portant sur l agilité et l efficacité des jambes des animaux, à des techniques innovatrices pour construire divers types de mécanismes de marche. L étude des systèmes électromécaniques munis de jambes se justifie principalement par les avantages que ceux-ci présentent sur des terrains rugueux, pour l exploration terrestre, l exploitation minière et des forêts, pour l automatisation industrielle, le travail en terrain dangereux, les cultures agricoles, aussi bien que par ses avantages potentiels dans le développement et l expérimentation des prothèses [GAP01]. Les mécanismes bipèdes sont des systèmes naturellement instables. Le contrôle de ces mécanismes est un problème difficile et multidisciplinaire, considéré comme le moment crucial de la locomotion. En se basant sur les aspects dynamiques de la locomotion, nous pouvons classer les mécanismes bipèdes en marcheurs passifs, marcheurs statiques, marcheurs dynamiques ou marcheurs seulement dynamiques. Les marcheurs passifs utilisent la force de gravité comme source d énergie. Ils peuvent se déplacer sur un plan descendant simplement en transformant l énergie gravitationnelle en énergie cinétique. Chez les robots statiquement stables 1, la projection verticale du centre de gravité reste toujours dans la région convexe appelée polygone de support 2. La stratégie de locomotion des robots statiques consiste à planifier les mouvements du robot de telle façon qu ils gardent toujours la projection du barycentre dans le pied support, en produisant des forces d inertie négligeables. Ces robots peuvent stopper leurs mouvements à n importe quel instant du cycle de marche et maintenir l équilibre. L inconvénient principal de ce type de marche est la faible vitesse de la locomotion, et la nécessité de pieds de grands dimensions et actifs. Les robots dynamiquement stables 3 utilisent des forces dynamiques et une réalimentation pour maintenir le contrôle de leurs mouvements. Ce type de robots accomplit des mouvements rapides et naturels grâce au principe d équilibre dynamique, appliqué en premier lieu par [VJ69], qui ont introduit la notion de Point de Moment Zéro 4 (PMZ), utilisé ensuite par d autres chercheurs scientifiques, voir [VBSS90, Go99]. Avec cette méthode, tous les membres du robot se déplacent de façon coordonnée pour pouvoir garder le PMZ, qui prend en compte les forces d inertie et les forces de gravitation dans le pied d appui. Si le PMZ est dans la région d appui 5, alors le robot est considéré comme dynamiquement stable. Si ce n est pas le cas, et que le robot tourne autour d un point à l extérieur de la région d appui, le pied d appui aura tendance à être soulevé de 1 Robots qui peuvent se déplacer seulement avec des positions définies de stabilité à chaque instant de leur démarche, c est-à-dire qui ne sont jamais dans une configuration instable de marche. 2 Région formée par les points de contact des pieds sur la terre. 3 Robots qui se déplacent avec des positions instables, et qui ont besoin d ajuster et d organiser intelligemment leurs mouvements pour rester stable à tout temps donné. 4 Point sur la terre autour duquel le moment de gravité et la force d inertie deviennent zéro. 5 C est-à-dire que le robot tourne toujours autour d un point dans la région d appui.

126 A.1. STABILISATION ORBITALE DE SYSTÈMES SOUS-ACTIONNÉS 83 terre ou pressé sur la terre, menant aussi à l instabilité. Ce critère de stabilité ne peut pas être appliqué à un robot qui n a pas d articulations aux chevilles. Les marcheurs purement dynamiques se meuvent sans équilibre statique ou dynamique. Ce genre de robot bipède a des pieds passifs ou aucun pied du tout, et utilise quelquefois une structure externe de support pour le balancement latéral. A.1.1 La Stabilité Orbitale L étude des oscillations est d un grand intérêt à maints égards. Il est fort intéressant de recherche chez les robots bipèdes, des mouvements périodiques. Cependant, la théorie de Stabilité de Lyapunov ne peut pas être directement appliquée à l analyse de la stabilité des robots marcheurs. En effet, si x(t) est la solution périodique d un robot autonome pur (trajectoire), et x(t + δ) une autre solution, pour chaque valeur de δ, les solutions périodiques d un système autonome ne peuvent pas être asymptotiquement stables de façon habituelle [GEK97]. Par conséquent, il convenient de définir la stabilité de ces types de systèmes par rapport à leur stabilité orbitale [HM86, Ha85]. Définition. Démarche Stable : Une démarche est stable si au démarrage sur un point à l intérieur de la trajectoire de phase C d un système (représenté par l Eq. (C.6)), une perturbation momentanée, quelle qu elle soit, produit une autre trajectoire proche de forme semblable. Définition. Démarche Stable Asymptotiquement : Une démarche est stable asymptotiquement si, en plus d être stable, le système revient au cycle original malgré les perturbations. Définition. Trajectoire Stable Orbitalement : La trajectoire de la phase C d un système (représenté par l Eq. (C.6)), est stable orbitalement si connaissant ɛ > 0, il existe δ > 0 tel que si R est un point représentatif d une autre trajectoire C qui se trouve à une distance δ de C à l instant t 0, alors C reste à la distance ɛ de C pour t 0. Si δ n existe pas, C est instable orbitalement. Définition. Trajectoire Stable Orbitalement Asymptotiquement : Une trajectoire C est stable orbitalement asymptotiquement si la trajectoire C est stable orbitalement, et si la distance entre C et C tend vers zéro quand le temps tend vers l infini. A.1.2 Le Problème du Balancement Les robots bipèdes appartiennent à une sous-classe de robots marcheurs [GAP01]. Leur caractéristique principale est le contact intermittent avec la terre, qui leur permet plus de souplesse dans leurs déplacements, mais dont résulte une instabilité structurelle de ces systèmes. Le balancement statique dans une démarche exige que le centre de masse soit sur la base d appui à tous moments. La plupart des robots marcheurs et des robots

127 84 APPENDIX A. RÉSUMÉ coureurs essaie d agir en faisant des mouvements périodiques (c est-à-dire des phases de balancement, de marche, de course, etc.). Ils requièrent un balancement actif pour pouvoir se déplacer quelque soit le type de trajectoire choisie, ce qui impliquera toujours que le centre de masse se trouve à l extérieur de la région d appui. Le problème du balancement stable peut être considéré comme une problématique subsidiaire intéressante à considérer avant ou après le déplacement du robot. Cette problématique a d ailleurs motivé notre travail Etude sur le problème du contrôle de la stabilisation orbitale d un robot plan à sept degré de liberté. Dans cette étude, nous considérons un prototype nommé Rabbit, voir Fig. A.1. Ce robot est un bipède plan sous-actionné à sept degrés de liberté, avec cinq liens rigides connectés l un à l autre par articulations rotationnelles, plus les coordonnées cartésiennes des hanches. Rabbit a un tronc et deux jambes avec genou, mais sans pieds. Les axes entre le tronc et chaque fémur sont actionnés, ainsi que les axes de chaque genou, mais le tronc n est pas actionné. Le contact entre la jambe d appui et la terre est un pivot. Ainsi, le contact entre la jambe et le sol est ponctuel, ce qui fait un modèle mathématique à cinq degrés de liberté. Figure A.1: Rabbit. Le robot marcheur à 5-DDL.

128 A.1. STABILISATION ORBITALE DE SYSTÈMES SOUS-ACTIONNÉS 85 A.1.3 Modèle Général Nous considérons une catégorie de systèmes électromécaniques sous-actionnés non holonomiques à n-dof avec des articulations non flexibles, où les forces généralisées sont des couples délivrés par les actionneurs, c est-à-dire que le système est conservateur en l absence d entrées extérieures, qui se réduiront d ailleurs pour nous aux couples de commande actionneurs. La dynamique de ce système est donnée par l équation de Lagrange suivante : M(q) q + C( q, q) q + g(q) = B τ (A.1) où, q R n est le vecteur de position des articulations, M(q) R nxn est la matrice de l inertie, C(q, q) R ( nxn) représente les forces centrifuges et la force de Coriolis, g(q) R n est le vecteur de gravité, B est une matrice constante de rang m, et τ R m est le vecteur de commande qui inclut toutes les forces généralisées externes, avec m < n comme nombre d actionneurs. La matrice B est définie de la façon suivante : B = [ 0(n m, m) I m ] (A.2) Posons l équation N(q, q) = C( q, q) q+g(q), et supposons que l espace d état puisse être transformé et divisé par un difféomorphisme φ en une partie à dimension m et une partie à dimension (n m); l espace de configuration q peut alors se diviser en deux ensembles de coordonnées (q a, q p ), et (A.1) peut de ce fait être réécrite de la façon suivante : avec : M p (q) q + N p (q, q) = 0 M a (q) q + N a (q, q) = τ q = [ qp q a ] (A.3) (A.4) (A.5) M(q) = [ Mp (q) M a (q) ] (A.6) N(q, q) = [ Np (q, q) N a (q, q) ] = [ Cp (q, q) + g p (q) C a (q, q) + g a (q) ] (A.7) où q p R n m et q a R m sont les angles de l articulation passive et de l articulation active. Dans l ensemble, le sous-indice p est utilisé pour représenter les quantités relatives à l articulation passive, alors que a représente les quantités relatives aux articulations actives. L Eq. (A.3) montre que le moment de rotation de l articulation passive est zéro, et représente un ensemble de (n m) contraintes sur le système, exprimées par des équations différentielles de second ordre, c est-à-dire que ce sont des contraintes imposées

129 86 APPENDIX A. RÉSUMÉ sur les accélérations angulaires généralisées pour toute commande d actionnement. Ces équations incluent les coordonnées généralisées q, les vitesses q, et les accélérations q. L Eq. (A.4) décrit la dynamique par rapport au moment de rotation τ R m des articulations actives. A.1.4 Objectif de Contrôle L objectif de contrôle est d amener le système (A.1) à la dynamique zéro et de le stabiliser, étant donné que les orbites périodiques stables de la dynamique zéro correspondent aux orbites stabilisables du système entier (A.1). Nous pouvons définir un ensemble de sorties du système telles que l annulation de ces sorties permette d obtenir la configuration souhaitée du robot. Par exemple, nous définissons la sortie à m dimensions suivante : y(q, t) := q a φ(q p, θ(t)) (A.8) où φ(q p, θ) est une fonction lisse, et θ est un vecteur paramétrique variant avec le temps, qui deviendra le contrôle utilisé pour imposer au système une dynamique zéro particulière. Comme dim(q a ) = dim(τ), toutes les forces/moments de rotation d actionnement disponibles doivent être utilisées dans ce but. Lorsque nous définissons que J(q) = y, q que d autre part nous faisons les manipulations suivantes à partir de (A.1) : q + M 1 N = M 1 B τ J q + J M 1 N = J M 1 B τ τ = (J M 1 B) 1 [J M 1 N + J q ] (A.9) (A.10) (A.11) et qu enfin nous supposons que JM 1 B n est pas singulier, au moins localement, la loi de réalimentation suivante : [ τ = (J M 1 B) 1 u + J M 1 N J ] q γ (A.12) avec 6 γ = J θ +J θ, linéarise la sortie y, c est-à-dire ÿ = u. Par conséquent, la linéarisation partielle au moyen de la solidarisation/réalimentation de (A.1) peut être obtenue en utilisant la loi de contrôle (A.12). Nous pouvons considérer beaucoup de structures de réalimentation pour s assurer que y 0 de façon asymptotique ou en un temps fini 7 ; ainsi, une grande famille de fonctions de réalimentation non-lisses peut être également conçue pour conduire le système à la dynamique zéro. Dans cette étude nous avons choisi la famille de fonctions de réalimentation non-lisses suivante [Ka96] : u = λ ẏ k s s k s n sign(s) s = ẏ + λ y ẏ = q p + J q + φ θ θ (A.13) (A.14) (A.15) 6 γ c est un autre terme impliqué dans la différentiation de (A.8), qui, introduit dans (A.1) donne ÿ = u. 7 La convergence est nécessaire pour éviter les problèmes dus à la fuite en temps fini, et par conséquent pour assurer le stabilité interne du système.

130 A.1. STABILISATION ORBITALE DE SYSTÈMES SOUS-ACTIONNÉS 87 où 0 n < 1, k et k s ont valeurs positifs. A.1.5 Dynamique Zéro La dynamique (A.1) projetée sur la contrainte y(q, t) = 0 (contrainte impliquant qu ẏ = ÿ = 0 et u = 0), est appelée dynamique zéro associée avec φ. Après avoir completé (A.1) à l aide de (A.11), et en faisant les manipulations suivantes : M q + N B (J M 1 B) 1 [J M 1 N + J q ] = 0 M q + N B (J M 1 B) 1 J M 1 [M q + N] = 0 (A.16) (A.17) [ In B (J M 1 B) 1 J M 1] [M q + N] = 0 (A.18) on peut obtenir cette dynamique zéro (Eq. (A.18)). En posant : P (q) := [ I n B (J M 1 B) 1 J M 1] (A.19) on peut réécrire (A.18) de la façon suivante : P (q) [M(q) q + N(q, q)] = 0 (A.20) où P (q) est l opérateur de projection sur le noyau de J M 1 dans la direction orthogonale de B. La dynamique zéro (A.20) repose sur un espace de configuration de dimension n m. A.1.6 Caractère Unique de la Dynamique Zéro Nous tirons de l Eq. (A.1) que : [ ] q q =: M 1 (q) + [ C(q, q) q g(q) + B τ] (A.21) En différentiant deux fois la sortie du système, nous tirons de (A.21) que : ÿ = L 2 f h(q, q) + L g L f h(q) u (A.22) Le fait d inverser la matrice L g L f h(q) à un point donné assure l existence et le caractère unique de la dynamique zéro dans le voisinage de ce point [Is95].

131 88 APPENDIX A. RÉSUMÉ A.1.7 Solutions Périodiques Stables Notre objectif est de trouver une fonction lisse φ et une loi d adaptation pour θ au moyen d une réalimentation dynamique, de telle sorte que la dynamique zéro présente des solutions périodiques. Si nous considérons des systèmes où le nombre des actionneurs est m = n 1, nous aurons une dynamique zéro de dimension 1 (dim(q p ) = 1), décrite par une équation non linéaire de deuxième ordre. Son comportement peut alors être étudié sur le plan. Si en outre nous considérons un système 2-DDL, et prenons une contrainte linéaire de la forme q a = aq p + b(t), ainsi que les contraintes suivantes : q a = φ(q p, θ) q a = J p (q p, θ) q p, +J θ (q p, θ) θ q a = J p (q p, θ) q p + r p (q p, q p, θ, θ, θ) (A.25) (A.23) (A.24) où : θ = θ = [ a b [ 0ḃ ] ] (A.26) (A.27) θ = [ 0 b ] (A.28) J p = φ q p J θ = φ θ r p = J p q p + J θ θ + J θ θ (A.29) (A.30) (A.31) en prenant (n m) lignes indépendantes du système suivant, alors la dynamique (A.20) peut être écrite en fonction des articulations non actionnées q p seulement. [ [ ] ] Jp q P (φ(q p ), q p, θ) M(φ(q p ), q p, θ) p + r p + N(φ(q q p ), q p, J p, q p, J θ, θ, θ) = 0 (A.32) p En définissant de la façon suivante : [ q T p ] z = (A.33) q T p

132 A.1. STABILISATION ORBITALE DE SYSTÈMES SOUS-ACTIONNÉS 89 nous pouvons représenter la dynamique zéro (A.32) par un ensemble approprié de coordonnées locales, à savoir : ż = f(z, φ(z, θ), θ, θ, θ) (A.34) Dans cette représentation de l espace d état, le vecteur θ nous donne 1 degré de liberté supplémentaire, utilisé pour générer un comportement oscillatoire stable dans (A.34). Si les éléments de θ ont une dynamique constante ou périodique, dans l Eq. (A.34), on peut imposer un mouvement périodique sur z au moyen de la restriction y = 0. Alors, la dynamique zéro agit comme un oscillateur autonome non linéaire qui conduit le reste des coordonnées du système à une orbite périodique, au moyen de la contrainte imposée par les coordonnées de l articulation actionnée q a = φ(q p, θ). Dans ce cas, l équation (A.32) prend la forme spécifique suivante : β 2 (q p, φ, θ) q p + β 0 (q p, q p, φ, θ, θ, θ) = 0 (A.35) q p O p Q P, Q P étant l espace de travail pour la variable non actionnée, et θ φ, où θ résulte de la loi d adaptation θ) la solution de (A.35) est bien définie si : β 2 (q p, φ, θ) > 0 (A.36) Définissant q d p comme le centre d orbite (l équilibre désiré), x = [ x1 x 2 avec : x 1 = q p qp d ] (A.37) (A.38) et x 2 = q p, alors on peut écrire (A.35) de la façon suivante : β 2 (x 1, q d p, φ, θ) ẍ 1 + β 0 (x, q d p, φ, θ, θ, θ) = 0 (A.39) équation qui, lorsque l on suppose (A.36), a la représentation dans l espace d état suivante : x = f(x, θ, φ, θ, θ), où : ẋ 1 = x 2 (A.40) ẋ 2 = β(x, φ, θ, θ, θ) avec β = β 0 β 2. A.1.8 Générateur d Orbite Le problème à résoudre à présent est la définition de l orbite désirée généralisée O d (x), en définissant une trajectoire fermée dans le plan de la phase, et en posant également que

133 90 APPENDIX A. RÉSUMÉ O d (x) définit un ensemble invariant et attractif de la solution, au moins localement, du générateur d orbite généralisé suivant : ẋ 1 = x 2 ẋ 2 = β d (x) (A.41) De cette façon, nous supposons que la fonction β d (x) arrive seulement à l équilibre quand x = 0 (β d (0) = 0), l équilibre étant inclus dans un ensemble convexe où O d (x) se trouve. En d autres termes, il existe là un ensemble fermé M R 2, qui ne contient pas de points d équilibre et est invariant positif. L orbite O d (x) définie semi-positive est contenue dans M. A.1.9 Dynamique Zéro Forcée Une fois la dynamique zéro obtenue, son mouvement dirigé par (A.20) est libre puisqu aucune autre commande n est disponible. Dans de nombreux cas, ce mouvement libre est une orbite périodique [ABEDM01]. Si on considère que le système (A.1) a 1 degré de sous-actionnement (m = n 1), alors sa dynamique zéro peut être exprimée en utilisant une seule coordonnée notée x 1. Si le portrait de phase de ce système est une courbe fermée O, alors cette orbite périodique qui caractérise la dynamique zéro peut être clairement identifiée comme étant unique grâce au triplet (φ, x 1, x 2 ). Il s agit à présent de définir une telle orbite périodique en tant qu objectif final, en rappelant que nous pouvons considérer le choix de φ comme une façon de modifier cet orbite. Pour ce faire, nous considérons la famille centrée d orbites elliptiques suivantes 8 : { O d = x : V d = 1 } 2 (α d x x 2 2) (A.42) comme étant une fonction des ensembles de paramètres {V d, α d, q d p}, dans laquelle : V d est le niveau désiré d orbite, α d est la forme désirée d orbite, et q d p est le centre désiré d orbite. Ces orbites attirent les solutions du générateur généralisé d orbite (A.41), β d (x) étant défini de la façon suivante : β d (x) = α d x 1 + k V x 2 Ṽ (x) (A.43) Dans cette équation, le niveau d énergie de l orbite, noté suivante : Ṽ (x), est défini de la façon Ṽ (x) = V (x) V d = 1 2 (α d x x 2 2) V d (A.44) c est-à-dire : ẋ 1 = x 2 ẋ 2 = α d x 1 k V x 2 [ ] 1 2 (α d x x 2 2) V d (A.45) 8 Au cause du changement des coordonées x 1 = q p q d p, les orbites de la fonction de Lyapunov V associées au système seront centrées autour de x 1 = 0.

134 A.1. STABILISATION ORBITALE DE SYSTÈMES SOUS-ACTIONNÉS 91 Pour vérifier cela, il faut définir v = Ṽ 2 /2, et noter que v = 2 k V x 2 2 v 0. Les deux seuls cas où v s annule sont les suivants : Quand la solution a atteint l orbite désirée (v = 0). Quand les conditions initiales sont prises au point d équilibre x = 0. Le cas précédent montre l invariance positive de O d, conséquence du fait que l orbite est centrée au point d équilibre (q d p, 0). A.1.10 Obtention de O d A partir d une orbite initiale donnée (en partant des conditions initiales données), le système (A.1) peut arriver à une orbite périodique spécifique O d, grâce à une loi de commande pour sa dynamique zéro et de là, à une dynamique particulière de la partie non commandée du système. Théorème 1. Soit θ 1 = θ, θ 2 = θ, Θ = [θ 1, θ 2 ] T. Considérons la dynamique zéro étendue suivante (avec φ = φ(x 1, Θ)) : ẋ 1 = x 2 ẋ 2 = β(x, φ, Θ, k(x, φ, Θ)) θ 1 = θ 2 θ 2 = k(x, φ, Θ) (A.46) où k(x, φ, Θ) définit la loi d adaptation de θ. être trouvé avec les propriétés suivantes : Supposons qu un k(x, φ, Θ) puisse 1. Une orbite désirée est définie par l équation (A.41), grâce à la définition d un β d (x) particulier. 2. Il existe là une fonction lisse φ(x 1, Θ), et un ensemble O Θ de conditions initiales appropriées pour Θ(0) = Θ 0, telles que : 2.1 β 2 (x 1 (t), φ(x 1 (t), Θ(t)), Θ(t)) > β(x(t), φ(x 1 (t), Θ(t)), Θ(t)) = β d (x) pour tout Θ(t), x 1 (t), t 0 résultant de la solution de (A.46), avec x(0) 0, et Θ(0) O Θ. 3. Le sous-système résultant, avec x (t) = x (t + T ), et x (t) <, θ 1 = θ 2 θ 2 = k(x, φ, Θ) (A.47)

135 92 APPENDIX A. RÉSUMÉ produit des solutions bornées. Pour tout x(0) 0 la solution x(t) converge alors à l orbite désirée.

136 A.1. STABILISATION ORBITALE DE SYSTÈMES SOUS-ACTIONNÉS 93 A.1.11 Conclusions Les systèmes électromécaniques sous-actionnés à n-ddl avec des contraintes non holonomiques que nous avons présentés, bien qu habituellement considérés comme tout à fait non linéaires ou bien caractérisés par un couplage non linéaire fort, peuvent se transformer en des systèmes non linéaires sur lesquels il sera facile de travailler, et une méthodologie de contrôle linéaire peut leur être appliquée avec succès. Pour produire des oscillations stables sur le système, il faut produire des cycles limites stables, étant donné que ce type d oscillations est associé à un cycle limite. Contrôler cette catégorie de systèmes électromécaniques sous-actionnés devient beaucoup plus difficile que dans le cas des systèmes holonomiques. Le système étudié appartient à une catégorie de systèmes appelés systèmes non linéaires de phase non minimale, qui jusqu à aujourd hui n a pas reçu de traitement théorique satisfaisant.

137 94 APPENDIX A. RÉSUMÉ A.2 Etude d un Modèle à Trois Articulations A titre d illustration, nous présentons une application du système de contrôle présenté dans la section précédente, sur un robot bipède pendant la phase de balancement. Cet exemple montre comment réduire l analyse de la stabilité d un système dynamique à 1-DDL, et donne les conditions générales pour assurer le comportement oscillatoire avec une stabilité interne. Etant donné qu un robot bipède peut être représenté par un système de pendule inversé, nous considérons le problème de la stabilisation orbitale d un pendule inversé à 3-DDL. Dans ce pendule inversé de haute dimension, la première articulation n a pas d entrée directe de contrôle, et le contact avec la terre est répresenté par un pivot. L objectif du contrôle est de trouver une loi de réalimentation telle que le pendule inversé soit capable d avoir un mouvement périodique de façon stable. Nous constations qu en changeant la période et/ou l amplitude du mouvement, nous pouvons avoir des sauts d un cycle limite valable à un autre cycle limite valable, tous deux étant appropiés. Pour produire un comportement oscillatoire dans tout l ensemble du système coordonné et sous-actionné, en supposant que la sortie m-dimensionnelle appropriée ait la forme donnée par l Eq. (2.8) (y(q, t) := q a φ(q p, θ(t))), avec q = [ ] qp T, qa T T, la méthodologie présentée dans la section précédente peut être résumée par les phases suivantes : 1. Il faut choisir une fonction de sortie appropriée, 2. Linéariser la sortie, 3. Etudier les conditions sous lesquelles cette sortie (y(q, t)) peut créer un cycle limite stable dans la dynamique zéro résultante, 4. Appliquer la stabilisation par une réalimentation sur le système du pendule inversé en équilibre instable. Par exemple, nous pouvons prendre une contrainte linéaire de la forme : q a = a q p + b(t) (A.48) où a R (m, n m) = R (m, 1) est un vecteur constant, b(t) R m est un vecteur qui varie avec le temps. En particulier, si a = (a 1, a 2,... a m ) T, alors le vecteur θ(t) = [a T, b(t) T ] T (A.49) caractérise les paramètres qui seront réglés (ou adaptés).

138 A.2. ETUDE D UN MODÈLE À TROIS ARTICULATIONS 95 A.2.1 Modèle de Robot à Trois Articulations : Le Robot Gymnaste Nous allons considérer l exemple d un robot plan sous-actionné à trois articulations. Ce robot, représenté Fig. A.2, est libre de tourner dans le plan vertical. Considérons les trois équations suivantes qui décrivent la dynamique de ce robot [ATS98] : m 11 q p + m 12 q a1 + m 13 q a2 + c 11 q p + c 12 q a1 + c 13 q a2 + g 1 = 0 m 21 q p + m 22 q a1 + m 23 q a2 + c 21 q p + c 22 q a1 + c 23 q a2 + g 2 = τ a1 m 31 q p + m 32 q a1 + m 33 q a2 + c 31 q p + c 32 q a1 + c 33 q a2 + g 3 = τ a2 (A.50) (A.51) (A.52) où les valeurs de m ij (q a1, q a2 ), c ij (q a1, q a2, q p, q a1, q a2 ), et g i (q p, q a1, q a2 ), i, j = 1,...3. ont été développés spécialement pour cette thèse, et peuvent être trouvées dans l Appendice D. Les sous-indices a1 et a2 sont utilisés pour représenter des quantités relatives aux articulations actives, alors que p représente les quantités relatives à l articulation passive. L Eq. (A.50) représente une contrainte dynamique qui ne peut pas être intégrée pour obtenir une relation algébrique entre les coordonnées q p, q a1, y q a2. - τ a2 - q a2 m 3 l c3 l 3 m 2 τ a1 l 2 l c2 q a1 m 1 q p l c1 l 1 Figure A.2: Robot plan à 3 articulations. Pour les simulations du robot, nous avons supposé les paramètres physiques suivants : Tableau A.1: Paramètres du Robot Gymnaste. l 1 = 0.47 [m] l c1 = 0.14 [m] m 1 = 1.48 [kg] I 1 = [kg m/s 2 ] l 2 = 0.26 [m] l c2 = 0.12 [m] m 2 = 1.06 [kg] I 2 = [kg m/s 2 ] l 3 = 0.29 [m] l c3 = 0.25 [m] m 3 = 0.29 [kg] I 3 = [kg m/s 2 ]

139 96 APPENDIX A. RÉSUMÉ Le système (A.50)-(A.52) projeté sur la contrainte y = ẏ = ÿ = 0, nous donne la dynamique zéro. La dynamique zéro résultante peut être calculée en remplaçant ces contraintes dans la partie non actionnée du système. La dynamique zéro peut être paramétrée grâce au choix d un type de fonction de sortie. Pour le système (A.50)-(A.52), nous définissons la sortie suivante de dimension 2 : [ ] qa1 a y(q, t) := 1 q p b 1 (t) (A.53) q a2 a 2 q p b 2 (t) A.2.2 Dynamique Zéro La complexité et la dimension de la dynamique zéro augmentent si les deux paramètres b i (t) varient avec le temps. Pour simplifier, nous considérerons seulement b1(t) comme variant avec le temps, tandis que b 2, a 1, et a 2 auront des valeurs constantes. On peut donc définir le vecteur de paramètres ainsi : θ = [a 1, a 2, b 1 (t), b 2 ] T. Ce choix particulier donne les résultats suivants : [ ] [ ] qa1 a1 q = p + b 1 (t) (A.54) q a2 a 2 q p + b 2 [ ] [ qa1 a1 q = p + ḃ1(t) ] (A.55) q a2 a 2 q p [ ] [ qa1 a1 q = p + b ] 1 (t) (A.56) q a2 a 2 q p et en les remplaçant dans (A.50), la dynamique zéro résultante prend la forme suivante : β 2 (q p, φ, θ) q p + β 0 (q p, q p, φ, θ, θ, θ) = 0 (A.57) où la solution sera bien définie si β 2 (q p, θ) > 0. Pour simplifier la notation des équations ci-dessous, nous ôterons le temps dans les différents arguments. A.2.3 Dynamique Zéro Forcée La méthode de contrôle introduite consiste à trouver une loi de contrôle qui oblige la dynamique zéro (A.57) à produire un mouvement cyclique stable. Cela peut être fait comme suit : nous définissons x 1 = q p q d p, et x 2 = q p, où q d p est le centre de l orbite désirée, et à partir de (A.57), nous tirons : ẋ 1 = x 2 ẋ 2 = β(x, φ, θ, θ, θ) (A.58)

140 A.2. ETUDE D UN MODÈLE À TROIS ARTICULATIONS 97 En considérant la famille d orbites elliptiques désirées donnée par V d = 1 2 (α dx x 2 2), et caractérisée par les paramètres choisis, la dynamique zéro (A.58) est contrainte de se comporter en tant que générateur d orbites (A.41), à condition que nous puissions trouver une loi d adaptation pour θ, qui satisfasse l équation différentielle suivante : β(x, φ, θ, θ, θ) = β d (x), t 0 (A.59) En définissant θ 1 = b 1, θ 2 = θ 1, alors la loi d adaptation prend, dans notre cas particulier, la forme suivante : θ 1 = θ 2 1 θ 2 = (f m 12 (x 1,θ) 5(x 1, θ) θ2 2 + f 7 (x 1, θ) x 2 θ 2 + f 8 (x 1, θ) x g 1 (x 1, θ) + β 2 (x 1, θ) β d (x)) (A.60) Un choix convenable de a est nécessaire pour obtenir m 12 > 0. Cependant, à cause du terme quadratique sur θ, les conditions initiales doivent être sélectionnées de façon convenable afin d éviter la fuite en temps fini. Nous avons vu d autre part qu il existait un ensemble compact de conditions initiales telles que l équation (A.46) reste asymptotiquement bornée. A.2.4 Résultats des Simulations Nous avons programmé le robot comme un pendule inversé à trois articulations pour montrer la stabilité et la périodicitéde ses mouvements. Les figures A.3-A.11 montrent des exemples de mouvements où la trajectoire converge dans l orbite O d. Les valeurs des paramètres choisis sont les suivantes : Tableau A.2: Paramètres des orbites elliptiques. V d = [rad 2 /s 2 ] α d = [1/s 2 ] a 1 = 1 a 2 = 2 b 2 = [rad] q d p = π/2 [rad] k V = 100 Tableau A.3: Conditions initiales des orbites elliptiques. q d p(0) = [rad] q p (0) = 0.3 [rad/s] b 1 (0) = [rad] ḃ 1 (0) = 0 [rad/s] La figure A.3 montre l évolution de q p où la position initiale du Robot Gymnaste est en dehors de la position d équilibre instable q p (0) = Cet angle, mesuré entre l axe horizontal et la première articulation du pendule inversé, est négatif dans le sens de rotation des aiguilles d une montre. Après quelques secondes, le système a des oscillations stables autour du plan vertical.

141 98 APPENDIX A. RÉSUMÉ q p [deg] T [s] Figure A.3: Evolution de q p. L action dynamique de q a1 y q a2 est présentée dans les Figs. A.4 et A q a1 [deg] T [s] Figure A.4: Evolution de q a1 (t). Pour compenser la position initiale de ce pendule inversé à 3-DDL, la vitesse initiale de l articulation non actionnée est différente de zéro (0.3 [rad/s]). Les Figs. A.6 et A.7 montrent le rapport vitesse/temps des articulations actionnées.

142 A.2. ETUDE D UN MODÈLE À TROIS ARTICULATIONS q a2 [deg] T [s] Figure A.5: Evolution de q a2 (t) dq a1 /dt [rad/s] T [s] Figure A.6: Evolution de q a1.

143 100 APPENDIX A. RÉSUMÉ dq a2 /dt [rad/s] T [s] Figure A.7: Evolution de q a2. Les résultats de la simulation présentée dans la Fig. A.8 montrent que le portrait de phase du système converge à l orbite elliptique désirée O d (x) (x 1 (0), x 2 (0)) 0.2 x 2 [rad/s] O d x [rad] 1 Figure A.8: Portrait de Phase du Robot Gymnaste. Convergence sur une orbite elliptique désirée. La dynamique zéro, montrée dans les Figs. A.9 et A.10, converge sur un cycle limite stable.

144 A.2. ETUDE D UN MODÈLE À TROIS ARTICULATIONS b 1 [deg] T [s] Figure A.9: Evolution de b 1 (t) db 1 /dt [rad/s] T [s] Figure A.10: Evolution de ḃ1. La dynamique zéro de ce système est bornée, voir Fig. A.11.

145 102 APPENDIX A. RÉSUMÉ db 1 /dt [rad/s] b 1 [rad] T [s] Figure A.11: Evolution de b 1 (t) y ḃ1.

146 A.2. ETUDE D UN MODÈLE À TROIS ARTICULATIONS 103 A.2.5 Conclusions Nous avons montré qu il était possible d obtenir une stabilité orbitale dans un système à trois articulations fonctionnant comme un pendule inversé, même si celui-ci utilise un actionneur de moins que le nombre de ses DDL. Les résultats obtenus nous permettent de définir une loi d adaptation appropiée à systèmes de plus grandes dimensions, en utilisant la même méthodologie employée pour le Robot Gymnaste. Cependant, ce degré supplémentaire de liberté ne fournit pas de moyens supplémentaires pour améliorer les propriétés de stabilité du système. Néanmoins, ces résultats permettent d obtenir une structure du contrôle qui donne des propriétés de stabilité semblables, malgré l augmentation de la complexité du système. L étude montré par ailleurs qu il est possible de traiter des cas où la dimension du robot est plus grande encore. La recherche effectuée ci-dessus est motivée par la possibilité d appliquer cette technique de contrôle sur notre modèle dynamique : un robot bipède à 5-DOF.

147 104 APPENDIX A. RÉSUMÉ A.3 Commentaires sur les Résultats Expérimentaux sur le Robot Rabbit Le robot Rabbit est un robot plan qui a été construit dans un but de recherche. Ce robot nous permet de valider expérimentalement des outils théoriques et des méthodes de contrôle visant à produire des mouvements chez les robots bipèdes. Rabbit nous permet d étudier des problèmes intéressants tels que les cycles limites stables pendant la phase de balancement (c est-à-dire lorsque que le robot s appuie sur une seule jambe et maintient un balancement), pendant la marche et pendant la course. Le concept qui a motivé la construction de Rabbit était la création d un robot anthropomorphique le plus simple possible, mais gardant toutefois les propriétés nécessaires à la marche humaine [CL02]. Ce mécanisme a sept DDL : cinq membres rigides connectés l un à l autre par des articulations rotationnelles, et deux DDL associés au déplacement horizontal et vertical du centre de masse 9 dans le plan sagittal. Rabbit a un tronc et deux jambes avec genou, mais pas de pieds. Les axes entre le tronc et chaque fémur sont actionnés, ainsi que les axes de chaque genou, mais le tronc ne l est pas. Ce prototype a donc sept degrés de liberté et quatre degrés d actionnement. Le contact entre la jambe d appui et la terre est un pivot. Ainsi, le contact entre la jambe et le sol est ponctuel, ce qui fait un modèle mathématique à cinq degrés de liberté. L articulation virtuelle entre la jambe d appui et la terre est une articulation unilatérale parce qu il n y a pas de forces attirantes présentes, et passive parce qu elle est sous-actionnée ; aussi, Rabbit ne peut pas générer de couple sur la terre. Pour garantir la stabilité latérale, il y a une structure de support externe qui ne permet pas les mouvements dans les plans transversal ou frontal, voir Fig. A.12. Le mouvement de Rabbit est donc limité à une évolution sur le plan sagittal, puisque le robot est maintenu par une tige autour d un poteau central au moyens d une barre radiale 10, voir Fig. A.13. Le robot a un mouvement de rotation libre autour du poteau central et l extrémité inférieure de chacune de ses jambes est munie d une roulette 11. Ces roulettes sont utilisées pour réduire le frottement radial pendant le déplacement de la jambe d appui. 9 Les coordonnées cartésiennes des hanches. 10 Le rayon de la trajectoire circulaire est approximativement 3 m. 11 Les roulettes ont été construites à partir d un polymère raide pour absorber les impacts.

148 A.3. COMMENTAIRES SUR LES RÉSULTATS EXPÉRIMENTAUX SUR LE ROBOT RABBIT 105 Plan horizontal Plan frontal Plan sagittal Figure A.12: Plans de référence par rapport au corps humain. Il y a 3-DDL entre Rabbit et la barre radiale, à savoir une articulation rotationnelle entre le tronc et la barre radiale alignée sur les axes des hanches (mouvement de rotation), et une articulation universelle entre la colonne centrale et la barre radiale (mouvement horizontal et mouvement vertical). Par conséquent, ces trois DDL passifs permettent la mobilité complète du Rabbit sur le plan sagittal. Figure A.13: Colonne centrale et barre radiale du Rabbit. La hauteur de ce robot est de m et sa masse est approximativement de 32 kg. La masse du fémur est de 6.8 kg (y compris le demi-réducteur de la hanche, le moteur du genou et le demi-réducteur du genou), la masse du tibia est de 3.2 kg (y compris le demi-réducteur genou-pied) et la masse du tronc est approximativement de 12 kg. Sa géométrie est détaillée dans la Fig. A.14.

149 106 APPENDIX A. RÉSUMÉ Tableau A.4: Paramètres physiques de Rabbit. Tronc Fémur Tibia Longueur [m] M asse [kg] Centre de masse [m] yg = 0.01; zg = 0.2 zg = zg = Inertie [kg m 2 ] Iox = 2.22 Iox = 0.25 Iox = 0.10 P remiers moments [kg m] MY = 0.2; MZ = 4.0 MZ = 1.11 MZ = 0.41 Inertie corrigée [kg m 2 ] Iox = 1.08 Iox = 0.93 Tableau A.5: Paramètres de l inertie des réducteurs de Rabbit. Irotor [kg m 2 ] Iréducteur [kg m 2 ] Irotor + Iréducteur [kg m 2 ] 2.25e e e-4 Tronc Moteur et codeur incrémental (hanche) Réducteur et codeur absolu (hanche) ) Moteur et codeur incrémental (genoux) Réducteur et codeur absolu (genoux) Figure A.14: Géométrie de Rabbit. Rabbit a été construit en aluminium pour réduire son poids et garantir una résistance déterminée. Tous les composants électriques et mécaniques sont montés de la façon la plus symétriquement possible pour garder le centre de masse autour des lignes de symétrie du robot.

150 A.3. COMMENTAIRES SUR LES RÉSULTATS EXPÉRIMENTAUX SUR LE ROBOT RABBIT 107 Dans le Tableau A.6 sont représentés les débattements articulaires de Rabbit. Tableau A.6: Association entre les débattements articulaires de Rabbit. Articulation Degrés (arrière) Degrés (avant) Barre/colonne centrale (hanche) autour de Sans limite de rotation Sans limite de rotation l axe horizontal Barre/colonne centrale (hanche) autour de l axe vertical Barre/T ronc (hanche) T ronc/f émur (hanche) - 80 (extension) 80 (f lexion) F émur/t ibia (genou) - 50 (f lexion) 50 (extension) A.3.1 La Boucle Interne Il est nécessaire de comprendre que du point de vue du contrôle automatique, il y a des difficultés importantes pour contrôler et stabiliser ce robot [CAAPWCG02] : Comme le système est sous-actionné, nous ne pouvons pas espérer commander tous les DDL sur des trajectoires préalablement définies 12. Le contact entre la jambe d appui et la terre est ponctuel et il peut être rompu en présence de perturbations 13. Un modèle qui capture tous les phénomènes physiques est mathématiquement très complexe. Le système réel comporte beaucoup de différences qui sont autant de sources d erreurs par rapport au système Lagrangian idéalisé, à savoir : Le frottement dans les réducteurs des moteurs, ainsi que le frottement dans l articulation universelle de la colonne centrale (Tour qui supporte le système électronique et le système DSpace). Les dynamiques non considérées, telles que la flexibilité du câblage d alimentation, la torsion des éléments des articulations, et la flexibilité de la barre de contrepoids. Les imprécisions paramétriques dues aux inexactitudes des mesures de l inertie des articulations du robot et de l inertie supplémentaire de la colonne centrale. Les impacts non-rigides dus à l intéraction entre l extrémité inférieure de chaque jambe et le sol. Les imprécisions dues à la mensuration des positions des articulations, comme par exemple les échantillonnages, les quantifications, etc. 12 Dans notre travail, nous faisons jouer des articulations, actives et passive, du robot, en partant des conditions initiales données jusqu à l orbite désirée O d. 13 Nous supposons que dans le contact entre la jambe d appui et la surface de la terre il y a un frottement suffisant pour éviter le glissement.

151 108 APPENDIX A. RÉSUMÉ Les outils classiques de contrôle automatique s avérant inefficaces pour analyser la stabilité de Rabbit, les concepts de stabilité doivent être correctement adaptés au robot. Perturbations d d d d (q HD, q HG, q GD, q GG ) + _ Contrôleur Robot RABBIT + _ q = [q T, q HD, q HG, q GD, q GG ] (q HD, q HG, q GD, q GG ) Générateur d'orbites q q T d Paramètres du Rabbit V d αd K a 1 a 2 b 1 (0) b 2 Figure A.15: Diagramme du système de contrôle de Rabbit. avec : q T : Angle du tronc. q HD : Angle de la hanche droite. q GD : Angle du genou droit. q HG : Angle de la hanche gauche. q GG : Angle du genou gauche. A.3.2 Commentaires sur les Résultats Experimentaux Pour évaluer l efficacité du contrôleur orbital proposé, nous présentons dans cette section quelques résultats expérimentaux. Notre méthodologie a recours au couplage mécanique de l articulation passive et des articulations actives en commandant les articulations actives afin de conduire l articulation passive sur l orbite désirée. Le générateur de trajectoires a été conçu de façon à générer des balancements sur Rabbit, sans spécifier les trajectoires de référence pour les articulations. Notre but est d amener le robot Rabbit à un état d oscillation au moyen de notre loi de commande, et d arriver à une courbe fermée qui produise un comportement oscillatoire stable. Une façon simple de représenter le mouvement d un bipède pendant la marche lente, avec un seul pied en contact avec la terre, est de considérer le modèle mathématique

152 A.3. COMMENTAIRES SUR LES RÉSULTATS EXPÉRIMENTAUX SUR LE ROBOT RABBIT 109 d un pendule inversé, voir Fig. A.16. Sur le schéma, le robot est représenté au moyen d un point de masse égal à la masse totale du robot, placé au centre de gravité, ayant une relation dynamique avec la terre grâce à une barre modelée mathématiquement sans masse 14. Figure A.16: Système de pendule inversé équivalent au robot. L objectif de cette expérience est de montrer que notre loi de commande est capable de stabiliser le système électromécanique 15 sur sa position d équilibre instable. Chaque fois que le centre de masse du robot traverse le point médian de la jambe dans une certaine direction, l autre jambe est activée pour résister au mouvement de chute du robot. Si le centre de gravité du robot n est pas projeté verticalement dans la région de la jambe, cela provoque une instabilité difficile à commander. Pendant la phase de l expérimentation, nous avons remarqué que la performance du contrôleur est sensible aux conditions initiales inexactes, les incertitudes paramétriques, et les forces du frottement. A cause de l attraction gravitationnelle, la région de l espace d état où le robot peut rester en équilibre est petite. Tous ces problèmes ont causé la chute du robot après quelques secondes de stabilité orbitale réussie. 14 Dans ce cas, le PMZ est le point de contact avec la terre. 15 Le robot Rabbit est représenté au moyen d un système de pendule inversé composé de liens multiples.

153 110 APPENDIX A. RÉSUMÉ A.3.3 Conclusions Dans cette section, la méthodologie de contrôle a été appliquée sur un robot plan nommé Rabbit. Le système étant sous-actionné, nous ne pouvons pas espérer commander tous les degrés de liberté sur des trajectoires préalablement définies. L existence d un degré non actionné dans l articulation virtuelle entre la jambe d appui et la terre joue un rôle important dans la stabilité du système complet. Pendant le balancement des robots bipèdes, il est toujours possible de trouver une force qui cause la chute du robot ; aussi, nous ne pouvons pas espérer que ce contrôleur puisse produire une stabilité globale dans tout le système. Il est nécessaire de considérer un contrôleur robuste, c est à dire qui puisse empêcher la chute du robot, malgré les incertitudes paramétriques et les forces de frottements.

154 A.4. CONCLUSION GÉNÉRALE 111 A.4 Conclusion Générale A.4.1 Conclusion Les systèmes électromécaniques sous-actionnés se sont révélés très utiles pour une large gamme d applications, comme par exemple pour les robots sous-marins, les robots utilisés dans l espace, les robots flexibles, les robots mobiles, et pour beaucoup d autres applications actuelles ou futures, en raison de leurs moindre poids et de leur moindre consommation d énergie, obtenus grâce à leur nombre inférieur d actionneurs. Néanmoins, ces systèmes ont un inconvénient notable: ils ont besoin de techniques de contrôle spéciales, parce que les lois de commande habituelles ne peuvent pas manier les complexités dynamiques et mathématiques résultantes. Cependant, les systèmes mécaniques complètement actionnées peuvent devenir des systèmes sous-actionnées si l un au moins des actionneurs ne fonctionne pas ; il est alors commode d avoir un système de contrôle pour les commander. Nous avons considéré le problème de la stabilisation orbitale d une catégorie de systèmes électromécaniques sous-actionnés avec un nombre de degrés de liberté égal au nombre d actionneurs moins un. Les conditions pour obtenir la convergence locale ont été établies grâce à un algorithme implicite spécialement désigné pour cette application et dont résulte une méthode de contrôle entièrement nouvelle. La méthode de contrôle que nous avons proposée contraint le système à arriver sur une orbite périodique isolée en produisant un mouvement oscillatoire stable sur l ensemble du pendule inversé et en faisant usage de l interaction dynamique entre l articulation passive et les articulations actives. Cependant, les cycles limite ne sont pas toujours valables du fait des contraintes imposées par la surface de la terre. La principale caractéristique de cette stratégie de contrôle, comparée à beaucoup d autres stratégies de boucle fermée proposées dans la littérature, est qu elle ne requière pas de références comme les trajectoires désirées. Le modèle mathématique initialement utilisé par le simulateur LAPIN pour décrire le comportement dynamique du robot Rabbit [Ro98] a dû être corrigé. Le nouveau modèle mathématique introduit dans ce travail et décrit dans l Appendice E, a été utilisé dans le simulateur LAPIN pour modeler la nouvelle loi de contrôle, et pour les expérimentations réalisées sur le robot Rabbit. Pour illustrer cette méthode, nous avons choisi des systèmes sous-actionnées comme le Robot Gymnaste et le Robot Rabbit. Nous avons présenté les résultats de plusieurs simulations qui démontrent l efficacité de la méthode de contrôle ; cependant, dans les essais expérimentaux, il est nécessaire de considérer un contrôleur qui marche convenablement malgré les incertitudes paramétriques

155 112 APPENDIX A. RÉSUMÉ et les forces de frottement, de façon à éviter la chute du robot. La méthode de contrôle nous permet de travailler avec des systèmes à plusieurs liens. Le travail qui a été présenté peut contribuer à résoudre des problèmes pratiques tels que l étude de la locomotion stable chez les humains qui utilisent des prothèses de jambe, et chez les robots bipèdes. Dans les simulations présentées, nous avons choisi des conditions initiales à l extérieur du cycle limite produit par les orbites elliptiques désirées O d ; cependant, des résultats de convergence similaires ont été observés quand on a pris des conditions initiales au-dedans des orbites. Nous avons décidé de travailler avec la dynamique zéro parce que nous avons eu besoin d une façon simple de déterminer la stabilité de la dynamique interne des systèmes non linéaires. Cependant, nous devons considérer que mme si la stabilité asymptotique locale de la dynamique zéro est suffisante pour garantir la stabilité asymptotique locale de la dynamique interne, on ne peut pas obtenir des résultats sur la stabilité globale ou sur la stabilité dans une région, et nous devons également considérer que la stabilité locale de la dynamique interne est seulement garantie si la dynamique zéro est exponentiellement globalement stable. A.4.2 Recherches Futures Les études futures viseront à déterminer de façon plus précise les paramètres du système, ainsi que des simulations utilisant ces paramètres ; en effet, la complexité de telles dispositifs entrane nécessairement des niveaux d incertitude élevés (dynamiques non modelées), qui doivent tre considérés afin d obtenir des résultats expérimentaux plus proches du comportement désiré. Les études futures viseront à déterminer les effets des perturbations sur le système et à réfléchir sur le moyen de les supprimer. Nous avons vu que certains systèmes compliqués tel que les robots bipèdes se déplaçant sur trois dimensions n ont pas nécessairement besoin de systèmes de contrôle compliqués pour accomplir une tche désirée, parce que la méthodologie de contrôle présentée dans ce travail ouvre la porte à l analyse de systèmes plus complexes, comme on l a vu grce aux résultats obtenus dans cette thèse. Les robots bipèdes seront probablement utilisés pour la locomotion sur terrain rugueux, l exploration terrestre, l exploitation minière et des forts, l automatisation industrielle, pour des opérations dans des environnements hasardeux, pour les cultures, pour l élaboration et l expérimentation des prothèses, etc. Cependant, la conception et la construction de machines qui se déplacent de façon efficace, par la marche ou la course, restent toujours un défi. Il est donc nécessaire, dans le cadre de la recherche sur les robots bipèdes, de continuer à analyser le balancement actif des robots marcheurs.

156 Appendix B Resumen B.1 Estabilización Orbital de Sistemas Sub- Actuados A continuación presentamos nuestro trabajo de investigación sobre la estabilización orbital de sistemas electromecánicos de n-grados de libertad (GDL) con un grado de sub-actuación. La mayoría de los robots caminantes concebidos hasta ahora generan, por medio de motores localizados en los tobillos, el torque necesario para permitir su caminata. Sin embargo, debido a esta arquitectura, es necesario que estos robots tengan grandes pies. Para evitar este problema, que descalifica la caminata en terrenos accidentados, una solución es poder balancear el sistema por medio de un actuador en la rodilla y dejar el tobillo sin accionamiento. Esto posibilitaría la caminata de un robot bípedo en diversos tipos de terrenos (incluso accidentados), debido al apoyo puntual de la pierna sobre el espacio de trabajo. Esta ha sido la motivación que origina nuestro estudio del problema de la estabilización orbital de sistemas sub-actuados. En los robots sub-actuados, debido a la presencia de articulaciones pasivas, su dinámica es considerablemente más compleja comparada con los sistemas completamente actuados. Esta presencia de articulaciones pasivas se justifica principalmente cuando: Un robot logra ser controlado a pesar de la falla de actuadores. Se puede utilizar menos energía para hacerlo funcionar i.e., para propósitos de ahorro de energía. Algunas articulaciones son intencionalmente pasivas por seguridad humana. Estos sistemas electromecánicos no holonómicos se manifiestan de múltiples formas a través de sus variadas aplicaciones, tales como, robots sub-marinos, robots para aplicaciones espaciales, robots flexibles, robots móviles y otras máquinas que imitan la locomoción animal, como los robots caminantes. Generalmente, los sistemas electromecánicos sub-actuados tienen puntos de equilibrio que dependen tanto de sus parámetros cinemáticos como dinámicos [DMO00]. 113

157 114 APPENDIX B. RESUMEN La caminata es una de las acciones humanas más comunes, la cual requiere un complejo sistema de control para poder lograr un eficaz desplazamiento. La caminata humana por ser una actividad principalmente pasiva, requiere un pequeño control activo [Mc90]. La locomoción por medio de pies ha sido empleada como mecanismo de transporte biológico durante millones de años. Desde hace muchos años los investigadores han combinado observaciones científicas, tales como de la agilidad y eficacia de las patas de los animales, con innovadora ingeniería para construir diversos tipos de mecanismos caminantes. El estudio de sistemas electromecánicos con patas está motivado principalmente por sus beneficios en la locomoción sobre terrenos irregulares, la exploración terrestre, la minería, la silvicultura, la automatización industrial, su empleo en ambientes peligrosos y el cultivo, así como sus potenciales beneficios en la concepción y ensayo de prótesis [GAP01]. Los mecanismos bípedos son sistemas naturalmente inestables. El control de estos mecanismos es un problema difícil y multidiciplinario, el cual es considerado como el aspecto crucial de una exitosa locomoción. Basándonos en los aspectos dinámicos de la locomoción resultante, podemos clasificar los mecanismos bípedos como pasivos, estáticos, dinámicos o caminantes dinámicos puros. Los caminantes pasivos usan la fuerza de gravedad de la tierra como fuente de energía, pudiendo desplazarse sólo en planos inclinandos convirtiendo la energía gravitacional en energía cinética. En un robot estáticamente estable 1, la proyección vertical del centro de gravedad siempre permanece dentro de la región convexa llamada polígono de apoyo 2. La estrategia de locomoción de los robots estáticos consiste en planificar los movimientos del robot de manera que estos siempre mantengan la proyección del baricentro dentro del pie de apoyo, dando lugar a fuerzas inerciales despreciables. Estos robots pueden detener el movimiento en cualquier instante del ciclo de marcha y mantener el equilibrio. La principal desventaja de este tipo de locomoción es su reducida velocidad y la necesidad de pies grandes y actuados. Los robots dinámicamente estables 3 utilizan fuerzas dinámicas y realimentación para mantener el control de sus movimientos. Este tipo de robots logra un movimiento rápido y natural por medio del principio de equilibrio dinámico, que fue aplicado por primera vez en [VJ69] con la introducción del Punto de Momento Cero 4 (PMC), subsecuentemente empleado por otros investigadores científicos, ver [VBSS90, Go99]. Con este método, que considera las fuerzas inerciales y gravitacionales, los diversos miembros del robot se mueven de una forma coordinada para poder mantener el PMC dentro del pie de apoyo. Si el PMC se encuentra en la región de apoyo, esto significa que el robot siempre está rotando alrededor de un punto dentro de la región de apoyo, luego el robot es considerado dinámicamente estable. Si este no es el caso, y el robot rota alrededor de un punto fuera 1 Robots que sólo pueden desplazarse a través de posiciones definidas de estabilidad durante toda la caminata, por lo tanto nunca presentan una configuración inestable. 2 Región formada por los puntos de contacto de los pies sobre la tierra. 3 Robots que en sus ciclos de marcha se mueven a través de posiciones inestables, necesitando planificar de una manera adecuada sus movimientos para poder mantener el equilibrio en todo momento. 4 Punto sobre la tierra alrededor del cual el momento de gravedad y la fuerza inercial se hacen cero.

158 B.1. ESTABILIZACIÓN ORBITAL DE SISTEMAS SUB-ACTUADOS 115 de la región de apoyo, el pie de apoyo tenderá a perder el contacto con la tierra o a hacer presión sobre ella, con lo cual se presenta la inestabilidad. Este criterio de estabilidad no puede ser aplicado en un robot sin articulaciones en los tobillos. Los caminantes dinámicos puros logran su locomoción sin equilibrio ni estático ni dinámico. Esta clase de robots bípedos tienen pies pasivos o simplemente no poseen pies, y a veces utilizan una estructura de soporte externa para el balanceo lateral. B.1.1 Etabilidad Orbital El estudio y comprensión de oscilaciones toman gran importancia en muchas aplicaciones. Por ejemplo, en los robots bípedos muchas veces movimientos periódicos son deseados. Sin embargo, la teoría de Estabilidad de Lyapunov no puede ser aplicada directamente en el análisis de estabilidad de robots caminantes porque si, x(t) es una solución periódica de un robot autónomo puro (trayectoria), y x(t + δ) otra solución, para cada valor de δ, las soluciones periódicas de un sistema autónomo no pueden ser asintóticamente estables de la manera usual [GEK97]. Por lo tanto, es natural definir la estabilidad de este tipo de sistemas en términos de su estabilidad orbital [HM86, Ha85]. Definición. Caminata Estable: Una caminata es estable si partiendo en un punto perteneciente a la trayectoria de fase C de un sistema (representado por la Ec. (C.6)), cualquier perturbación finita induce otra trayectoria cercana de forma similar. Definición. Caminata Estable Asintóticamente: Una caminata es estable asintóticamente si es estable, y si a pesar de la presencia de perturbaciones el sistema regresa al ciclo original. Definición. Trajectoria Estable Orbitalmente: La trayectoria de fase C de un sistema (representado por la Ec. (C.6)) es estable orbitalmente si dado un ɛ > 0, existe un δ > 0 tal que si, R es un punto representativo de otra trayectoria C que está dentro de una distancia δ de C en un tiempo t 0, luego C permanece dentro de una distancia ɛ de C para t 0. Si este δ no existe, C es inestable orbitalmente. Definición. Trajectoria Estable Orbitalmente Asintóticamente: Una trajectoria C es estable orbitalmente asintóticamente si la trayectoria C es estable orbitalmente, y la distancia entre C y C tiende a cero cuando el tiempo tiende al infinito. B.1.2 El Problema de Balanceo Los robots bípedos pertenecen a una sub clase de robots caminates [GAP01]. Su pincipal característica es el contacto intermitente con la tierra, permitiendoles mayor versatilidad en sus desplazamientos, pero resultando en una inestabilidad estructural de estos sistemas. En una caminata, el balanceo estático requiere que el centro de masa se encuentre sobre la base de apoyo en todo momento. La mayoría de los robots caminantes

159 116 APPENDIX B. RESUMEN y robots corredores tratan de funcionar bajo movimientos periódicos (i.e., fases de balanceos, caminata, carrera, etc.). Ellos requieren un balanceo activo para poder desplazarse no importando que tipo de trayectoria haya sido escogida, esto siempre implicará que el centro de masa quede fuera del área de apoyo. El problema de balanceo estable puede ser considerado como una sub tarea interesante a ser abordada a priori o a posteriori del desplazamiento del robot. Esto ha motivado nuestro trabajo El problema del control de estabilización orbital de un robot plano de siete grados de libertad. En este trabajo, nosotros consideramos un prototipo llamado Rabbit, ver Fig. B.1. Este robot es modelado como un robot bípedo plano sub-actuado, que tiene 7-GDL, cinco eslabones rígidos conectados entre si por medio de articulaciones rotacionales, más las coordenadas cartesianas de las caderas. Rabbit tiene un tronco y dos piernas, pero no tiene pies. Los ejes entre el tronco y cada femur están actuados, como también los ejes de cada rodilla, pero el tronco no está actuado. El contacto entre la pierna de apoyo y la tierra es modelado como un pivote, luego el contacto entre la pierna y el suelo es puntual, y los GDL del modelo se reducen a cinco. Figura B.1: Rabbit. El robot caminante de 5-GDL. B.1.3 Modelo General Nosotros consideramos una clase de sistemas electromecánicos sub-actuados de n-gdl con restricciones no holonómicas, donde las fuerzas generalizadas son accionamientos de tipo torques/fuerzas solamente, sin flexibilidades y donde no se producen otros tipos de interacciones. La dinámica de este sistema es dada por medio de la siguiente ecuación de

160 B.1. ESTABILIZACIÓN ORBITAL DE SISTEMAS SUB-ACTUADOS 117 Lagrange: M(q) q + C( q, q) q + g(q) = B τ (B.1) donde q R n es el vector de posición de articulaciones, M(q) R nxn es la matriz de inercia, C(q, q) R ( nxn) es la matriz de efectos de Coriolis y de torques centrífugos, g(q) R n es el vector de gravedad, B es una matriz constante de rango m, y τ R m es el vector de torques que incluye todas las fuerzas externas generalizadas, con m < n el número de actuadores. La matriz B es definida como: B = [ 0(n m, m) I m ] (B.2) Sea N(q, q) = C( q, q) q + g(q), y asumiendo que el espacio-estado puede ser transformado y dividido por medio de un difeomorfismo φ en una parte m-dimensional y una parte (n m)-dimensional; luego el espacio configuración q puede ser dividido en dos conjuntos de coordenadas (q a, q p ), por lo tanto (B.1) puede ser expresada como: M p (q) q + N p (q, q) = 0 M a (q) q + N a (q, q) = τ (B.3) (B.4) con: q = [ qp q a ] (B.5) M(q) = [ Mp (q) M a (q) ] (B.6) N(q, q) = [ Np (q, q) N a (q, q) ] = [ Cp (q, q) + g p (q) C a (q, q) + g a (q) ] (B.7) donde q p R n m y q a R m corresponden a los ángulos de la articulación pasiva y las articulaciones activas, respectivamente. En general, el subíndice p es empleado para denotar las cantidades relativas a la articulación pasiva, y el subíndice a representa esto para las articulaciones activas. La Ec. (B.3) muestra que el torque de la articulación pasiva es nulo, y representa un conjunto de (n m) restricciones en el sistema, que son expresadas por ecuaciones diferenciales de segundo orden, i.e., restricciones impuestas sobre las aceleraciones angulares generalizadas admisibles para cualquier tipo de controlador. Estas restricciones incluyen las coordenadas generalizadas q, las velocidades q, y las aceleraciones q. La Ec. (B.4) describe la dinámica con respecto al torque τ R m de las articulaciones activas.

161 118 APPENDIX B. RESUMEN B.1.4 Objetivo del Control El objetivo del control es llevar el sistema (B.1) a su dinámica cero, y luego lograr estabilizarla considerando que las órbitas periódicas estables de la dinámica cero corresponden a las órbitas estabilizables de todo el sistema dinámico modelado en (B.1). Nosotros podemos especificar un conjunto adecuado de salidas, de tal manera que hacer cero estas salidas sea equivalente a obtener la configuración deseada del robot. Para este sistema, nosotros definimos la siguiente salida m-dimensional: y(q, t) := q a φ(q p, θ(t)) (B.8) donde φ(q p, θ) es una función suave, y θ es un vector de parámetros variable, quien constituirá el control usado para obtener una dinámica cero determinada. Como la dim(q a ) = dim(τ), todas las fuerzas/torques de accionamiento disponibles tienen que ser usadas para este propósito. Definiendo J(q) = y, haciendo las siguientes manipulaciones q a partir de (B.1): q + M 1 N = M 1 B τ J q + J M 1 N = J M 1 B τ τ = (J M 1 B) 1 [J M 1 N + J q ] (B.9) (B.10) (B.11) y asumiendo que JM 1 B no es singular, por lo menos localmente, la siguiente ley de realimentación: [ τ = (J M 1 B) 1 u + J M 1 N J ] q γ (B.12) con 5 γ = J θ +J θ, linealiza la salida y i.e., ÿ = u. Por consiguiente, la linealización parcial por medio del desacoplamiento/realimentacion de (B.1), puede ser realizada usando el control (B.12). Nosotros podemos tomar muchas estructuras de realimentación de salidas para asegurar que y 0 asintóticamente o en tiempo finito 6, por lo tanto, una gran familia de funciones no suaves de realimentación pueden ser diseñadas para conducir el sistema a la dinámica cero. En este estudio escogimos la siguiente familia de funciones no suaves de realimentación [Ka96]: u = λ ẏ k s s k s n sign(s) s = ẏ + λ y ẏ = q p + J q + φ θ θ donde 0 n < 1, k y k s son constantes positivas. (B.13) (B.14) (B.15) 5 γ Es otro termino involucrado en la diferenciación de (B.8), que introducido en (B.1) produce ÿ = u. 6 La convergencia es necesaria para evitar los problemas debido al escape en tiempo finito, y por lo tanto para asegurar la estabilidad interna del sistema.

162 B.1. ESTABILIZACIÓN ORBITAL DE SISTEMAS SUB-ACTUADOS 119 B.1.5 Dinámica Cero La dinámica cero se define como la dinámica interna resultante del sistema cuando las condiciones iniciales requeridas y los controles son aplicados para mantener en cero las salidas del sistema durante todo tiempo t [HY99]. La dinámica cero de (B.1), i.e., (B.3), también puede ser vista como una restricción no holonómica asociada a la parte accionada del sistema. La dinámica (B.1) proyectada sobre la restricción y(q, t) = 0, que implica ẏ = ÿ = 0 y u = 0, es llamada la dinámica cero asociada a φ. Después de sustituir (B.11) en (B.1), y haciendo las siguientes manipulaciones: M q + N B (J M 1 B) 1 [J M 1 N + J q ] = 0 M q + N B (J M 1 B) 1 J M 1 [M q + N] = 0 (B.16) (B.17) [ In B (J M 1 B) 1 J M 1] [M q + N] = 0 (B.18) nosotros podemos obtener esta dinámica cero (Eq. (B.18)). Definiendo: P (q) := [ I n B (J M 1 B) 1 J M 1] (B.19) nosotros podemos reescribir (B.18) como: P (q) [M(q) q + N(q, q)] = 0 (B.20) donde P (q) es el operador proyección sobre el nucleo de J M 1 en la dirección ortogonal de B. La dinámica cero (B.20) cae entonces en un espacio de configuración de dimensión n m. B.1.6 Unicidad de la Dinámica Cero Desde la Ec. (B.1) nosotros tenemos: [ ] q q =: M 1 (q) + [ C(q, q) q g(q) + B τ] (B.21) Diferenciando dos veces la salida, desde (B.21) se obtiene: ÿ = L 2 f h(q, q) + L g L f h(q) u (B.22) La invertibilidad de la matriz de acoplamiento L g L f h(q) en un punto dado asegura la existencia y unicidad de la dinámica cero en la vecindad de ese punto [Is95].

163 120 APPENDIX B. RESUMEN B.1.7 Soluciones Periódicas Estables Nuestro objetivo es encontrar una función suave φ y una ley de adaptación para θ por medio de una realimentacion dinámica, tal que la dinámica cero presente soluciones periódicas estables. Si nosotros consideramos sistemas donde el número de actuadores es m = n 1, nosotros tendremos una dinámica cero de dimensión uno (dim(q p ) = 1) descrita por una ecuación no lineal de segundo orden. Luego su comportamiento podrá ser estudiado en el plano. Si nosotros consideramos un sistema de 2-GDL, una restricción de tipo lineal de la forma q a = aq p + b(t), y las siguientes restricciones: q a = φ(q p, θ) q a = J p (q p, θ) q p, +J θ (q p, θ) θ q a = J p (q p, θ) q p + r p (q p, q p, θ, θ, θ) (B.25) (B.23) (B.24) donde: θ = θ = [ a b [ 0ḃ ] ] (B.26) (B.27) θ = [ 0 b ] (B.28) J p = φ q p J θ = φ θ r p = J p q p + J θ θ + J θ θ, (B.29) (B.30) (B.31) y tomando (n m) líneas independientes del sistema siguiente, la dinámica (B.20) puede ser escrita como una función que sólo dependa de las articulaciones no accionadas q p. [ [ ] ] Jp q P (φ(q p ), q p, θ) M(φ(q p ), q p, θ) p + r p + N(φ(q q p ), q p, J p, q p, J θ, θ, θ) = 0 (B.32) p Definiendo: [ q T p ] z = (B.33) q T p nosotros podemos representar la dinámica cero (B.32) por medio de un adecuado conjunto de coordenadas locales, i.e., ż = f(z, φ(z, θ), θ, θ, θ) (B.34)

164 B.1. ESTABILIZACIÓN ORBITAL DE SISTEMAS SUB-ACTUADOS 121 En esta representación espacio-estado, el vector θ nos da un grado de libertad adicional, que es usado para producir un comportamiento oscilatorio estable en (B.34). Si en la Ec. (B.34), los elementos de θ tienen una dinámica constante o periódica, nosotros podemos imponer un movimiento periódico en z por medio de la restricción impuesta por y = 0. Luego, la dinámica cero actúa como un oscilador no lineal autónomo que conduce el resto de las coordenadas del sistema a una órbita periódica, por medio de la restricción impuesta por las coordenadas de las articulaciones accionadas q a = φ(q p, θ). En este caso, la ecuación (B.32) toma la forma particular: β 2 (q p, φ, θ) q p + β 0 (q p, q p, φ, θ, θ, θ) = 0 (B.35) Si q p O p Q P, donde Q P es el espacio de trabajo de la variable no accionada, y si θ φ, donde θ resulta de la ley de adaptación θ, entonces la solución de (B.35) estará bien definida, i.e., β 2 (q p, φ, θ) > 0 (B.36) Definiendo qp d como el centro de la órbita (equilibrio deseado), [ ] x1 x = con: x 1 = q p qp d x 2 (B.37) (B.38) y x 2 = q p, entonces nosotros podemos escribir (B.35) como: β 2 (x 1, q d p, φ, θ) ẍ 1 + β 0 (x, q d p, φ, θ, θ, θ) = 0 (B.39) que bajo la suposición (B.36) tiene la siguiente representación espacio-estado x = f(x, θ, φ, θ, θ): ẋ 1 = x 2 (B.40) ẋ 2 = β(x, φ, θ, θ, θ) con β = β 0 β 2. B.1.8 Generador de Orbitas A continuación introduciremos una órbita deseada generalizada O d (x), definiendo un camino cerrado en el plano de fase, y considerando además que O d (x) define un conjunto invariante y atractivo de la solución, por lo menos localmente, del siguiente generador de órbitas generalizado: ẋ 1 = x 2 ẋ 2 = β d (x) (B.41)

165 122 APPENDIX B. RESUMEN Por lo tanto, asumimos que la función β d (x) solamente llega al equilibrio cuando x = 0 (β d (0) = 0), equilibrio incluído en un conjunto convexo donde se encuentra O d (x). En otros términos, aquí existe un conjunto cerrado M R 2, luego M no contiene ningún punto de equilibrio y es invariante positivo. La órbita O d (x) definida semi-positiva entonces está contenida en M. B.1.9 Dinámica Cero Forzada Una vez alcanzada la dinámica cero, su movimiento gobernado por (B.20) es libre ya que no se dispone de ningun otro tipo de control. En muchos casos, este movimiento libre es una órbita periódica [ABEDM01]. Si nosotros consideramos que el sistema (B.1) tiene un grado de sub-actuación (m = n 1), entonces su dinámica cero puede expresarse usando una sola coordenada denotada por x 1. Si el plano de fase de este sistema es una curva cerrada O (órbita), entonces esta órbita periódica, que caracteriza la dinámica cero, puede ser especificada de manera única por el trío (φ, x 1, x 2 ). Nuestro objetivo final es alcanzar esta órbita periódica por medio de una adecuada elección de φ como una manera de modificar las órbitas. Para ello consideramos la siguiente familia de órbitas elípticas deseadas 7 : { O d = x : V d = 1 } 2 (α d x x 2 2) (B.42) como una función del conjunto de parámetros {V d, α d, q d p}, donde: V d es el nivel deseado de la órbita, α d es la forma deseada de la órbita, y q d p es el centro deseado de la órbita. Estas órbitas atraen las soluciones del generador de órbitas generalizado (B.41), con β d (x) definido como: β d (x) = α d x 1 + k V x 2 Ṽ (x) (B.43) donde el nivel de energía de la órbita, denotado por forma: Ṽ (x), es definido de la siguiente Ṽ (x) = V (x) V d = 1 2 (α d x x 2 2) V d (B.44) es decir: ẋ 1 = x 2 ẋ 2 = α d x 1 k V x 2 [ ] 1 2 (α d x x 2 2) V d (B.45) Para ver esto, definimos v = Ṽ 2 /2, y notamos que v = 2 k V x 2 2 v 0. Los 7 Debido al cambio de coordenadas x 1 = q p q d p, las órbitas de la funcion de Lyapunov V asociadas al sistema serán centradas alrededor de x 1 = 0.

166 B.1. ESTABILIZACIÓN ORBITAL DE SISTEMAS SUB-ACTUADOS 123 únicos dos casos donde v es igual a cero se producen cuando: La solución alcanza la órbita deseada (v = 0). las condiciones iniciales se toman en el equilibrio x = 0. El caso anterior muestra la invarianza positiva de O d como consecuencia de que la órbita está centrada en el punto de equilibrio (q d p, 0). B.1.10 Obtención de O d Dada una órbita inicial (partiendo de condiciones iniciales dadas), el sistema (B.1) puede alcanzar una órbita periódica específica O d, a través de una ley de control para su dinámica cero, produciendo una dinámica particular en la parte no accionada del sistema. Teorema 1. Definiendo θ 1 = θ, θ 2 = θ, Θ = [θ 1, θ 2 ] T. Considerando la siguiente dinámica cero extendida (con φ = φ(x 1, Θ)): ẋ 1 = x 2 ẋ 2 = β(x, φ, Θ, k(x, φ, Θ)) θ 1 = θ 2 θ 2 = k(x, φ, Θ) (B.46) donde k(x, φ, Θ) define la ley de adaptación para θ. Asumiendo que podemos encontrar un k(x, φ, Θ) tal, que cumpla con las siguientes propiedades: 1. Una órbita deseada es definida por el conjunto de ecuaciones (B.41), a lo largo de la definición de un β d (x) determinado. 2. Ahí existe una función suave φ(x 1, Θ), y un conjunto O Θ de condiciones iniciales apropiadas para Θ(0) = Θ 0, tales que: 2.1 β 2 (x 1 (t), φ(x 1 (t), Θ(t)), Θ(t)) > 0, 2.2 β(x(t), φ(x 1 (t), Θ(t)), Θ(t)) = β d (x) para todo Θ(t), x 1 (t), t 0 resultante de la solución de (B.46), con x(0) 0, y Θ(0) O Θ. 3. El subsistema resultante, con x (t) = x (t + T ), y x (t) <, θ 1 = θ 2 θ 2 = k(x, φ, Θ) (B.47) produce soluciones acotadas.

167 124 APPENDIX B. RESUMEN Entonces, para todo x(0) 0 la solución x(t) converge a la órbita deseada.

168 B.1. ESTABILIZACIÓN ORBITAL DE SISTEMAS SUB-ACTUADOS 125 B.1.11 Conclusiones Los sistemas electromecánicos sub-actuados de n-gdl con restricciones no holonómicas que hemos considerado en nuestro estudio, que normalmente son considerados altamente no lineales y caracterizados por fuertes acoplamientos no lineales, pueden ser transformados en sistemas no lineales fácilmente trabajables, de modo que una metodología de control lineal pueda ser empleada. Para generar oscilaciones estables en todo el sistema, es necesario que se produzcan ciclos límites estables, ya que este tipo de oscilaciones están asociadas a un ciclo límite. Controlar esta clase de sistemas electromecánicos sub-actuados es mucho mas difícil que en el caso de sistemas holonómicos. El sistema estudiado pertenece a una clase de sistemas llamados no lineales de fase no mínima, que todavía no tiene tratamiento teórico satisfactorio.

169 126 APPENDIX B. RESUMEN B.2 Estudio de un Modelo de Tres Eslabones A modo de ilustración, a continuación presentamos una aplicación del sistema de control descrito en la sección precedente, sobre un robot bípedo en la fase de balanceo. Este ejemplo muestra la manera de reducir el análisis de estabilidad de un sistema dinámico a 1-GDL, y da las condiciones generales para asegurar el comportamiento oscilatorio con estabilidad interna. Ya que un robot bípedo puede ser representado por un sistema de péndulo invertido, nosotros trabajamos el problema de la estabilización orbital de un péndulo invertido de 3-GDL. En este péndulo invertido de alta dimensión, la primera articulación no tiene una entrada directa de control y el contacto con la tierra es modelado como un pivote. El objetivo de control es encontrar una ley de realimentación tal que el péndulo invertido sea capaz de presentar un movimiento periódico de una manera estable. Veremos que cambiando el periodo y/o la amplitud del movimiento es posible generar saltos entre ciclos límites admisibles 8. Para generar un comportamiento oscilatorio en todo el conjunto del sistema coordenado sub-actuado, la metodología presentada en la sección precedente, asumiendo que la salida m-dimensional adecuada tiene la forma dada por la Ec. (B.8) (y(q, t) := q a φ(q p, θ(t))), con q = [ ] qp T, qa T T, puede ser resumida en las fases siguientes: 1. Es necesario escoger una función de salida adecuada, 2. Linealizar la salida, 3. Etudiar las condiciones bajo las cuales esta salida (y(q, t)) puede generar un ciclo límite estable en la dinámica cero resultante, 4. Aplicar estabilización por realimentación sobre el sistema de péndulo invertido en equilibrio inestable. Como ejemplo, podemos tomar una restricción lineal de la forma: q a = a q p + b(t) (B.48) donde a R (m, n m) = R (m, 1) es un vector constante, b(t) R m es un vector variable en el tiempo. En particular, si a = (a 1, a 2,... a m ) T, entonces el vector θ(t) = [a T, b(t) T ] T (B.49) caracteriza los parametros que serán ajustados (o adaptados). 8 Los ciclos límites admisibles no son completamente libres porque ellos deben respetar las restricciones impuestas por el espacio físico de trabajo y las fuerzas de contacto en el extremo de la pierna.

170 B.2. ESTUDIO DE UN MODELO DE TRES ESLABONES 127 B.2.1 Modelo del Robot de Tres Eslabones: El Robot Gimnasta Como ejemplo, consideramos un robot plano sub-actuado de tres eslabones como lo esquematiza la Fig. B.2, el cual puede rotar libremente en el plano vertical. Consideremos las ecuaciones que modelan este robot [ATS98]: m 11 q p + m 12 q a1 + m 13 q a2 + c 11 q p + c 12 q a1 + c 13 q a2 + g 1 = 0 m 21 q p + m 22 q a1 + m 23 q a2 + c 21 q p + c 22 q a1 + c 23 q a2 + g 2 = τ a1 m 31 q p + m 32 q a1 + m 33 q a2 + c 31 q p + c 32 q a1 + c 33 q a2 + g 3 = τ a2, (B.50) (B.51) (B.52) donde las expresiones exactas para m ij (q a1, q a2 ), c ij (q a1, q a2, q p, q a1, q a2 ), y g i (q p, q a1, q a2 ), i, j = 1,...3. fueron especialmente desarolladas para esta tesis, y pueden ser encontradas en el Apéndice D. Los subíndices a1 y a2 son usados para representar las cantidades relativas a las articulaciones activas, mientras p representa esto para la articulación pasiva. La Ec. (B.50) representa una restricción dinámica que no puede ser integrada para obtener una relación algebraica entre las coordenadas q p, q a1, y q a2. - τ a2 - q a2 m 3 l c3 l 3 m 2 τ a1 l 2 l c2 q a1 m 1 q p l c1 l 1 Figura B.2: Robot plano de tres eslabones. Los siguientes parámetros físicos del robot fueron considerados en los cálculos computacionales: Tabla B.1: Parámetros del robot Gimnasta. l 1 = 0.47 [m] l c1 = 0.14 [m] m 1 = 1.48 [kg] I 1 = [kg m/s 2 ] l 2 = 0.26 [m] l c2 = 0.12 [m] m 2 = 1.06 [kg] I 2 = [kg m/s 2 ] l 3 = 0.29 [m] l c3 = 0.25 [m] m 3 = 0.29 [kg] I 3 = [kg m/s 2 ]

171 128 APPENDIX B. RESUMEN El sistema (B.50)-(B.52) proyectado sobre la restriccion y = ẏ = ÿ = 0, nos entrega la dinámica cero. La dinámica cero resultante puede ser calculada sustituyendo estas restricciones en la parte no accionada del sistema. La dinámica cero puede ser parametrizada por medio de la elección de una clase de funciones de salida. Para el sistema (B.50)-(B.52), definimos la siguiente salida de dimensión 2: [ ] qa1 a y(q, t) := 1 q p b 1 (t) (B.53) q a2 a 2 q p b 2 (t) B.2.2 Dinámica Cero La complejidad y la dimensión de la dinámica cero aumentan si ambos parámetros b i (t) son variables en el tiempo. Por simplicidad de diseño, sólo b 1 (t) será considerado como variable, mientras que b 2, a 1, y a 2 son elementos con valores constantes. El vector de parámetros es entonces definido como θ = [a 1, a 2, b 1 (t), b 2 ] T. Esta elección particular entrega los siguientes resultados: [ ] [ ] qa1 a1 q = p + b 1 (t) (B.54) q a2 a 2 q p + b 2 [ ] [ qa1 a1 q = p + ḃ1(t) ] (B.55) q a2 a 2 q p [ ] [ qa1 a1 q = p + b ] 1 (t) (B.56) q a2 a 2 q p y sustituyendo en (B.50), la dinámica cero resultante tiene la forma: β 2 (q p, φ, θ) q p + β 0 (q p, q p, φ, θ, θ, θ) = 0 (B.57) cuya solución estará bien definida si β 2 (q p, θ) > 0. Sólo por simplicidad notacional, a continuación no escribiremos la dependencia del tiempo en los argumentos respectivos. B.2.3 Dinámica Cero Forzada El método de control descrito consiste en encontrar una ley de control que fuerce la dinámica cero (B.57) de manera que esta presente un movimiento cíclico estable. Esto puede lograrse como sigue: definiendo x 1 = q p q d p y x 2 = q p, donde q d p es el centro de la órbita deseada, y desde (B.57), nosotros tenemos: ẋ 1 = x 2 ẋ 2 = β(x, φ, θ, θ, θ) (B.58)

172 B.2. ESTUDIO DE UN MODELO DE TRES ESLABONES 129 Considerado la familia de órbitas elípticas deseada, dada por V d = 1 2 (α dx x 2 2), caracterizada por los parámetros de diseño, la dinámica cero (B.58) es forzada a comportarse de la misma forma que lo hace el generador de órbitas (B.41) si nosotros podemos encontrar una ley de adaptación θ, tal que la siguiente ecuación diferencial sea satisfecha: β(x, φ, θ, θ, θ) = β d (x), t 0 (B.59) Definiendo θ 1 = b 1, θ 2 = θ 1, entonces la ley de adaptación toma, en nuestro caso particular, la forma: θ 1 = θ 2 1 θ 2 = (f m 12 (x 1,θ) 5(x 1, θ) θ2 2 + f 7 (x 1, θ) x 2 θ 2 + f 8 (x 1, θ) x g 1 (x 1, θ) + β 2 (x 1, θ) β d (x)) (B.60) Una adecuada elección de a es necesaria para que m 12 > 0. Sin embargo, debido al término cuadrático en θ, las condiciones iniciales deben ser adecuadamente seleccionadas para evitar el escape en tiempo finito. Adicionalmente, hemos visto que existe un conjunto compacto de condiciones iniciales que permiten que la ecuación (B.46) permanezca asintóticamente acotada. B.2.4 Resultados de las Simulaciones El robot fue programado para mostrar estabilidad y periodicidad en su desempeño como péndulo invertido constituído por tres eslabones. Las figuras B.3-B.11 muestran ejemplos de movimiento donde la trayectoria converge a la órbita O d. Los valores de los parámetros escogidos son: Tabla B.2: Parámetros de las órbitas elípticas. V d = [rad 2 /s 2 ] α d = [1/s 2 ] a 1 = 1 a 2 = 2 b 2 = [rad] q d p = π/2 [rad] k V = 100 Tabla B.3: Condiciones iniciales de las órbitas elípticas. q d p(0) = [rad] q p (0) = 0.3 [rad/s] b 1 (0) = [rad] ḃ 1 (0) = 0 [rad/s] La figura B.3 muestra la evolución de q p donde la posición inicial del Robot Gimnasta está fuera de la posición de equilibrio inestable q p (0) = Este ángulo, medido entre el eje horizontal y el primer eslabón del péndulo invertido, es negativo en el sentido de las

173 130 APPENDIX B. RESUMEN agujas del reloj. Después de algunos segundos, el sistema presenta oscilaciones estables alrededor del plano vertical q p [deg] T [s] Figura B.3: Evolución de q p. La acción dinámica de q a1 y q a2 es presentada en las Figs. B.4 y B q a1 [deg] T [s] Figura B.4: Evolución de q a1 (t). Para compensar la posición inicial de este péndulo invertido de 3-GDL, la velocidad inicial del eslabón no actuado es diferente del cero (0.3 [rad/s]). Las Figs. B.6 y B.7 muestran la velocidad verus tiempo de los eslabones actuados.

174 B.2. ESTUDIO DE UN MODELO DE TRES ESLABONES q a2 [deg] T [s] Figura B.5: Evolución de q a2 (t) dq a1 /dt [rad/s] T [s] Figura B.6: Evolución de q a1.

175 132 APPENDIX B. RESUMEN dq a2 /dt [rad/s] T [s] Figura B.7: Evolución de q a2. Los resultados de la simulación presentada en la Fig. B.8 demuestran que el plano de fase del modelo en estudio converge a la órbita elíptica deseada O d (x) (x 1 (0), x 2 (0)) 0.2 x 2 [rad/s] O d x [rad] 1 Figura B.8: Plano de Fase del Robot Gimnasta. Convergencia a una órbita elíptica deseada. La dinámica cero, mostrada en las Figs. B.9 y B.10, converge a un ciclo límite estable.

176 B.2. ESTUDIO DE UN MODELO DE TRES ESLABONES b 1 [deg] T [s] Figura B.9: Evolución de b 1 (t) db 1 /dt [rad/s] T [s] Figura B.10: Evolución de ḃ1. La dinámica cero de este sistema está acotada, esto es mostrado en la Fig. B.11.

177 134 APPENDIX B. RESUMEN db 1 /dt [rad/s] b 1 [rad] T [s] Figura B.11: Evolución de b 1 (t) y ḃ1.

178 B.2. ESTUDIO DE UN MODELO DE TRES ESLABONES 135 B.2.5 Conclusiones Por medio de este estudio hemos podido mostrar que es posible obtener estabilidad orbital en un sistema compuesto de tres eslabones funcionando como un péndulo invertido, a pesar de emplear un actuador menos que el número de GDL del sistema. Los resultados obtenidos dan la posibilidad de tener una ley de adaptación de dimensión más alta, es decir, trabajar con sistemas de mayor dimensión empleando la misma metodología utilizada en el Robot Gimnasta. Sin embargo, este grado de libertad adicional no proporciona mayores posibilidades que permitan reforzar las propiedades de estabilidad del sistema. No obstante, esto produce una estructura de control que proporciona propiedades de estabilidad similares, a pesar del aumento de la complejidad del sistema. Por lo tanto, este estudio también muestra que es posible trabajar con casos donde la dimensión del robot es aun más grande. La motivación del trabajo realizado está basada en la posibilidad de aplicar esta técnica de control sobre nuestro modelo dinámico, un robot bípedo de 5-GDL.

179 136 APPENDIX B. RESUMEN B.3 Comentarios Sobre los Resultados Experimentales en el Robot Rabbit Rabbit es un robot plano que fue construído para propósitos de investigación. Este robot nos proporciona la posibilidad de validar, en forma experimental, herramientas teóricas y métodos de control para generar movimientos en robots bípedos. Rabbit nos permite estudiar interesantes problemas, tales como: ciclos límites estables durante la fase de balanceo (i.e., el robot se mantiene en balanceo sobre una sola pierna), durante la caminata y la carrera. La filosofía de concepción de Rabbit ha sido construir un robot antropomórfico lo más simple posible, pero que conserve las propiedades necesarias para que aún pueda permitir la representación de la caminata humana [CL02]. Este mecanismo tiene 7-GDL: cinco miembros rígidos conectados entre si sólo por articulaciones rotacionales, más 2-GDL asociados al desplazamiento horizontal y vertical de su centro de masa 9 en el plano sagital. Rabbit tiene un tronco y dos piernas con rodillas pero sin pies. Los ejes entre el tronco y cada femur están actuados, como también los ejes de cada rodilla, pero el tronco no está actuado. De este modo, este prototipo tiene siete grados de libertad y cuatro grados de actuación. El contacto entre la pierna de apoyo y la tierra es modelado como un pivote, luego el contacto entre la pierna y el suelo es puntual, y sus grados de libertad se reducen a cinco. La articulación virtual entre la pierna de apoyo y tierra es una articulación unilateral ya que no hay fuerzas atractivas presentes y pasiva porque ésta no está accionada, así Rabbit no puede aplicar torques sobre la tierra. Para garantizar estabilidad lateral, existe una estructura de soporte externo, que no permite movimientos en los planos transversal o frontal, ver Fig. B.12. De esta forma, el movimiento de Rabbit está limitado a una evolución en el plano sagital, por medio de una barra radial que gira alrededor de un puesto central 10, ver Fig. B.13. El robot tiene un movimiento libre alrededor de este puesto central y una pequeña rueda en el extremo inferior de cada pierna 11. Estas ruedas son usadas para permitir la menor fricción radial posible durante el desplazamiento de la pierna de apoyo. 9 Las coordenadas Cartesianas de las caderas. 10 El radio del la barra es de aproximádamente 3 m. 11 Las ruedas fueron fabricadas a partir de un polímero que permite absorber impactos.

180 B.3. COMENTARIOS SOBRE LOS RESULTADOS EXPERIMENTALES EN EL ROBOT RABBIT 137 Plano horizontal Plano frontal Plano sagital Figura B.12: Planos de referencia del cuerpo humano. Entre Rabbit y la barra radial hay 3-GDL; una articulación en revolución entre el tronco y la barra, la cual está alineada con el eje de las caderas (movimiento rotacional), más una articulación universal entre el puesto central y la barra (movimiento horizontal y vertical). De esta forma, estos 3-GDL pasivos le permiten al robot bípedo una completa movilidad sobre el plano sagital. Figura B.13: Puesto central y barra radial de Rabbit. Este robot mide m y pesa aproximadamente 32 kg. La masa del femur es de 6.8 kg (incluyendo la mitad del reductor de la cadera, el motor de la rodilla y la mitad del reductor de la rodilla), la masa de la tibia es de 3.2 kg (incluyendo la mitad del reductor de la rodilla), la masa del tronco es de aproximadamente 12 kg. Su geometría es detallada en la Fig. B.14.

181 138 APPENDIX B. RESUMEN Tabla B.4: Parámetros físicos de Rabbit. Tronco Femur Tibia Longitud [m] M asa [kg] Centro de masa [m] yg = 0.01; zg = 0.2 zg = zg = Inercia [kg m 2 ] Iox = 2.22 Iox = 0.25 Iox = 0.10 P rimeros momentos [kg m] MY = 0.2; MZ = 4.0 MZ = 1.11 MZ = 0.41 Inercia corregida [kg m 2 ] Iox = 1.08 Iox = 0.93 Tabla B.5: Parámetros de inercia de los reductores de Rabbit. Irotor [kg m 2 ] Ireductor [kg m 2 ] Irotor + Ireductor [kg m 2 ] 2.25e e e-4 Tronco Motor y encoder incremental (cadera) Reductor y encoder absoluto (cadera) Motor y encoder incremental (rodilla) Reductor y encoder absoluto (rodilla) Figura B.14: Geometría de Rabbit. Rabbit fue fabricado en aluminio para minimizar su peso y garantizar una determinada resistencia. Todos los componentes eléctricos y mecánicos están montados considerando la mayor simetría posible, para poder mantener el centro de masa a lo largo de las líneas de simetría del robot.

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