Modeling, stability and control of biped robots—a general framework

Abstract

The focus of this survey is the modeling and control of bipedal locomotion systems. More specifically, we seek to review the developments in the field within the framework of stability and control of systems subject to unilateral constraints. We place particular emphasis on three main issues that, in our view, form the underlying theory in the study of bipedal locomotion systems. Impact of the lower limbs with the walking surface and its effect on the walking dynamics was considered first. The key issue of multiple impacts is reviewed in detail. Next, we consider the dynamic stability of bipedal gait. We review the use of discrete maps in studying the stability of the closed orbits that represent the dynamics of a biped, which can be characterized as a hybrid system. Last, we consider the control schemes that have been used in regulating the motion of bipedal systems. We present an overview of the existing work and seek to identify the needed future developments. Due to the very large number of publications in the field, we made the choice to mainly focus on journal papers.

Introduction

In general, a bipedal locomotion system consists of several members that are interconnected with actuated joints. In essence, a man-made walking robot is nothing more than a robotic manipulator with a detachable and moving base. Design of bipedal robots has been largely influenced by the most sophisticated and versatile biped known to man, the man himself. Therefore, most of the models/machines developed bear a strong resemblance to the human body. Almost any model or machine can be characterized as having two lower limbs that are connected through a central member. Although the complexity of the system depends on the number of degrees of freedom, the existence of feet structures, upper limbs, etc., it is widely known that even extremely simple unactuated systems can generate ambulatory motion. A bipedal locomotion system can have a very simple structure with three point masses connected with massless links (Garcia, Chatterjee, Ruina, & Coleman, 1997) or very complex structure that mimics the human body (Vukobratovic, Borovac, Surla, & Stokic, 1990). In both cases, the system can walk several steps. The robotics community has been involved in the field of modeling and control of bipeds for many years. The books (Vukobratovic, 1976; Vukobratovic et al., 1990; Raibert, 1986; Todd, 1985) are worth reading as an introduction to the field. The interested reader may also refer to the following web pages:

http://www.androidworld.com/prod28.htm,

http://robby.caltech.edu/~kajita/bipedsite.html,

http://www.fzi.de/divisions/ipt/WMC/preface/preface.html,

http://www.kimura.is.uec.ac.jp/faculties/legged-robots.html.

Nevertheless, and despite the technological exploit achieved by Honda's engineers (Japan is certainly the country where bipedal locomotion has received the most attention and has the longest history), some fundamental modeling and control problems have still not been addressed nor solved in the related literature. One may notice, in particular, that the locomotion of Honda's P3 prototype remains far from classical human walking patterns at the same speeds. Although Honda (HONDA) did not publish many details either on the mechanical part or on the implemented control heuristic, it is easy to see on the available videos that P3's foot strike does not look natural and leads to some transient instability (http://www.honda-p3.com). The number of foot design patents taken out by Honda (up to an air-bag-like planter arch) reveals again that foot–ground impact remains one of the main difficulties one has to face in the design of robust control laws for walking robots. This will become the key issue with increasing horizontal velocity requirement. This problem, however, is more sensitive for two-legged robots than for multi-legged ones due to the almost straight leg configuration and the bigger load at impact time for the former, leading to stronger velocity jumps of the center of mass. While Honda's engineers seem to consider these velocity jumps as unwanted perturbations and thus appeal to mechanical astuteness to smooth the trajectory, we argue that impact is an intrinsic feature of mechanical systems like biped robots and should be taken as such in the controller design. Other bipedal robots have been designed. Among the most advanced projects, we cite the Waseda University Humanoid Robotics Institute biped, the MIT Leg Laboratory robots, the LMS-INRIA BIP system (Sardain, Rostami, & Bessonnet, 1998; Sardain, Rostami, Thomas, & Bessonnet, 1999), the CNRS-Rabbit project (Chevallereau et al., 2003), and the German Autonomous Walking programme (Gienger, Löffler, & Pfeiffer, 2003), which can be found at

http://www.humanoid.rise.waseda.ac.jp/booklet/kato_4.html,

http://www.ai.mit.edu/projects/leglab/robots/robots.html,

http://www.inrialpes.fr/bip,

http://www-lag.ensieg.inpg.fr/PRC-Bipedes/,

http://www.fzi.de/ids/dfg_schwerpunkt_laufen/start_page.html.

respectively. Among all these existing bipeds, the Honda robots seem to be the most advanced at the time of writing of this paper according to the information made available by the owners. However, the solution for control designed by Honda does not explain why a given trajectory works nor does it give any insight as to how to select, chain together, and blend various behaviors to effect locomotion through difficult terrain (Pratt, 2000). It is the feeling of the authors that the problem of feedback control of bipedal robots will not be solved properly as long as the dynamics of such systems is not thoroughly understood. In fact, the main motivation for the writing of this paper has been the following observation about walking: there is no analytical study of a stable controller with a complete stability proof available in the related literature. It is our belief that the main reason for this is the lack of a suitable model. We propose a framework that is not only simple enough to allow subsequent stability and control studies but also realistic as some experimental validations prove. In addition, the framework provides a unified modeling approach for mathematical, numerical, and control problems, which has been missing. It is for instance significant that the main efforts of the MIT Leg Lab (Pratt, 2000) have been directed toward technological (actuators) improvement and testing of heuristic control algorithms similar to Honda's works.

We should emphasize that the main thrust of this survey does overlook several practical aspects that may arise during the design and development of walking machines. Admittedly, a walking machine can be built without paying attention to many of the main ideas of this survey. There are numerous toys that walk in a certain fashion. There are quite a few bipedal robots that are designed to avoid impacts altogether during walking. The fact remains that the stability, agility, and versatility of any existing bipedal machine does not even come close to that of the human biped. The surveyed concepts will better enable the design and evaluation of such machines through more suitable control algorithms that take into account impact mechanics and stability. The practical issues that arise in the design and development actual machines deserve another survey article. In the ensuing part of this survey we, therefore, will mainly focus on a theoretical framework.