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Book Review Feedback Control of Dynamic Bipedal Robot Locomotion—Eric R. Westervelt, Jessy W. Grizzle, Christine Chevallereau, Jun Ho Choi, and Benjamin Morris (London, U.K.: Taylor & Francis, 2007). Reviewed by Jack B. Aldrich I. BACKGROUND Underlying the study of anthropomorphic or energy-efficient bipedal robot locomotion, is the challenging mathematical problem of determining the existence and stability of periodic solutions (e.g., walking/ running gaits) to impulsive dynamical systems (i.e., a specific class of hybrid systems that includes bipedal robots as a special case). Before attempting a solution to this problem, a natural place to start is to sample the massive number of publications regarding hybrid systems from the controls community. In doing so, one is quick to observe that many of the classical tools developed for nonlinear systems [5] (e.g., Lyapunov stability, invariance, dissipativity) have recently been extended to impulsive and hybrid dynamical systems ([3], and others). However, most of these results cannot be used to study bipedal locomotion as they pertain to the study of equilibrium points, not periodic orbits. To date, the only viable tool for studying periodic orbits in both hybrid and nonlinear systems has been the well-known Poincaré method. This method unfortunately has some serious limitations. For instance, in the ideal control design scenario, we would be able to parameterize the complete set of admissible (i.e., stable) periodic solutions a priori, so that optimization over an undefined admissible set would not be necessary. However, the Poincaré method does not, in general, lead to closed form conditions for the stability of periodic orbits. As a result of this shortcoming, most work on this problem has focused on designing energy-efficient locomotive gaits where the stability of the resulting periodic orbit is either ignored or assessed numerically after the design phase is complete [2]. Several years ago, however, the authors of the text under review were perceptive in recognizing that a closed-form test1 for the existence and stability of bipedal robot locomotive gaits was attainable. In particular, the authors carefully crafted a feedback control structure and a set of mild hypotheses that enabled them to make the following assertion: a bipedal robot is guaranteed to converge to a periodic orbit (e.g., user-defined gait) if and only if a pair of scalar inequality constraints is satisfied. In other words, the closed-loop stability assessment of a cleverly controlled bipedal robot within an -dimensional state-space no longer requires numerical integration of the Poincaré return map over an ( 0 1)-dimensional Poincaré section. To say the least, this result is theoretically interesting. Indeed, its mathematical validation is truly a multidisciplinary venture—one that draws substantially from geometric nonlinear control [4] and differential geometry, in addition to the major contributions from Poincaré analysis, and Lagrangian mechanics. Practically speaking, the result produces a stable, time-invariant feedback control law—an effective option for getting bipedal robots to walk and jog with robustness [1].
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The reviewer is with the Jet Propulsion Laboratory, Caltech, Pasadena, CA 91109 USA (e-mail:
[email protected]). Digital Object Identifier 10.1109/TAC.2008.918096 1The salient feature of a closed-form test is that a stability assessment is performed without time-integration of the system model. For instance, Lyapunov’s second method for equilibrium points is a closed-form test, but Poincaré’s method for periodic orbits is generally not.
In my assessment, the book under review is a well-focused, self-contained, and masterfully-written account of this result, which includes its mathematical foundations, validation, and practical implications. The authors have also done an excellent job encompassing the relevant abstractions of their approach. More specifically, they establish their stability results from both a hybrid systems framework (i.e., one not limited to bipedal robots), and a robotic systems framework, (i.e., one that exploits inherent structures within the Lagrangian mechanics formulation). As such, the elegant synergy between the feedback and the mechanics is gradually revealed over the course of several chapters. In my opinion, this dual perspective is well-organized and executed with great precision. The book will likely serve as a impetus for future work in hybrid systems research. II. THE BOOK In this monograph, Westervelt et al. present a general Poincaré-type analysis and design framework for systems with impulse effects and, in particular, to bipedal robots governed by a specific class of feedback controllers. Such a framework is built on the stability principals of invariance and attractivity, and the analytical formalisms from geometric nonlinear control theory and Lagrangian mechanics. The book is divided into three parts. The first part is preliminary and introduces the basic feedback control problem for bipedal robot locomotion. The second section develops the entire analysis and design framework for bipedal robots with point feet. The third section effortlessly adjusts the theory from section two to accommodate bipedal robots with nontrivial feet and actuated ankles. As such, the second section (chapters 4, 5, and 6, in particular) represents the core of the book in the sense that once these are mastered the rest of the book is easy to understand. For those that do not want to read the book from cover to cover, I highly recommend Appendix A: Getting Started, which is (as advertised) a great place to start. This Appendix asks the reader to follow a step-by-step procedure that effectively uses a simple example to guide the reader quickly though the essential information of the book. Specifically, the reader completes the following exercises: derivation of the hybrid model (i.e., bipedal dynamics and impact map), closed-loop simulation, numerical evaluation of Poincaré return maps, hybrid zero dynamics derivation/simulation, parameter optimization. Appendix B covers the fundamental notions and definitions from differential geometry, geometric nonlinear control, Poincaré analysis, and Lagrangian mechanics that are assumed throughout the text. As such, Westervelt et al. is self-contained. The eleven chapters within the book are organized as follows. Chapter 1 introduces the problem of controlling bipedal robot locomotion, and surveys much of the prior work on this subject. In particular, the design of energy-efficient walking and running gaits for biped robots is typically approached as a time-based trajectory optimization problem. For instance, in [2] and others, a finite-dimensional polynomial basis for the robot’s joint trajectories is assumed and an efficient gait is found that minimizes the expended input energy per unit step length subject to dynamic and unilateral constraints. As such, the closed-loop system is non-autonomous, and the stability analysis becomes problematic. In contrast, the text under review focuses on a feedback control structure that renders the closed-loop system time-invariant. Specifically, for a bipedal robot with -degrees-of-freedom, an under-actuated control approach is taken where the 0 1 actuated degrees of freedom are slaved to the evolution of the one unactuated
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degree of degree of freedom that is typically present at the ground/foot interface. The advantage of this approach is that the robot’s motion is “clocked” to itself, not an external function of time. The feasibility of this approach lies in the fact that the feedback variable is monotonic over each step taken by the biped robot. Chapter 2 introduces two biped robot testbeds that were used to evaluate the controllers proposed in the text. This chapter addresses practical design and implementation issues that arise when attempting to induce robustly stable bipedal robot walking gaits in a laboratory setting. Chapter 3 develops two continuous-time Lagrangian dynamic models—one models biped robots that are in point contact with the ground (i.e., single-support phase of the walking gait) and the other models bipeds in flight (i.e., flight phase of the running gait). In both running and walking gaits there is, of course, an impact phase and this is described by an algebraic impact model that is responsible for resetting the velocities of the joints immediately after impact according to the conservation of momentum law. Taken together, the impact model and the Lagrangian dynamics model form the hybrid system model that is treated extensively throughout the remainder of the text. Chapter 4 extends the classical Poincaré results for nonlinear systems [5] to impulsive dynamical systems. The proofs that establish these results are especially interesting because they do not assume Lipschitz continuity as is done in the nonlinear system case. It is worthwhile to note that the results of this important chapter can be applied to any hybrid system whose continuous dynamics are instantaneously reinitialized by a smooth (algebraic) impact map whenever a discrete-event (e.g., impact) criteria is satisfied. The chapter begins by assuming a minimal set of hypotheses (not limited to biped robots), and demonstrates how the Poincaré method can be applied to autonomous systems with impulse effects. As is the case for both hybrid and non-hybrid nonlinear systems, the Poincaré’s method reduces the stability assessment of a periodic orbit in a -dimensional state-space to the stability 0 1-dimensional discrete-time system. Although assessment of a this result is interesting in its own right, one often wonders what is the practical payoff in applying the Poincaré method to periodic systems due to the fact that the -dimensional state-space model still needs to be numerically integrated in general, before the discrete-time model can be constructed. As mentioned previously, the authors cleverly work around this liability by adopting a practical feedback control strategy that ensures the hybrid system dynamics are attractive (in some sense) to a smaller dimensional manifold , that is invariant (in some sense). In particular, when is two-dimensional, the domain of the Poincaré return map gets reduced from 0 1-dimensional down to the 1-dimensional \ , since traverses . Consequently, stability analysis becomes one-dimensional. The lesson here is that relegating the closed-loop system to a lower-dimensional manifold (i.e., via feedback or otherwise) increases the utility of the Poincaré approach. Depending on the type of attractivity and invariance hypotheses assumed, this low-dimensional stability test takes on different forms. For instance, finite-time attractive controllers can be designed to ensure the state of the closed-loop hybrid system reaches before any impacts occur, which ensures that the state enters the Poincaré section if and only if it enters \ , which follows due to the forward and impact invariance properties of . Alternatively, when the more standard Lipschitz continuous feedback control structures are assumed, it is no longer possible to assume the state will enter \ when an impact occurs. Nonetheless, a similar low-dimensional stability test is established by assuming the dynamics transversal to manifold are “sufficiently rapidly exponentially attractive” so that the classical singular perturbation (time-scale separation) argument can be applied. In summary, the results of chapter 4 are established for a general hybrid
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system class that is referred to in the literature as a “systems with impulsive effects,” or “impulsive dynamical systems”. Chapter 5 introduces the notion of “virtual constraints,” i.e., algebraic constraints that when driven to zero put the biped into its desired gait that evolves on the lower dimensional manifold . The chapter characterizes the two-dimensional zero dynamics that these constraints impose. Essentially, this chapter conditions the general results of chapter 4 to the more specific case of bipedal robots. In doing so, the Poincaré stability test no longer requires numerical integration, but rather, can be expressed in closed-form as a pair of scalar inequality constraints. An important application of this result, namely gait optimization, is investigated in chapter 6. However, the new feature here is that the existence and stability of walking and running gaits can now be guaranteed during the control design phase, rather than checked ex post facto. Specifically, the authors present a practical numerical optimization problem where a relevant cost function (e.g., expended-energy per bipedal step length) is minimized subject to several inequality constraints—one constraint establishes existence of a fixed-point, another establishes stability of the fixed-point, and yet another fixes the walking rate, etc. Moreover, the output functions (“virtual constraints”) are parameterized by a finite-dimensional polynomial basis, whose optimal values can be easily computed using Matlab’s nonlinear programming function, fmincon. Incidentally, the book includes several references to websites (managed by the authors), where downloads of the free simulation software are made available. Although the proposed optimization approach seems effective, the presentation fails to mention the fact that the problem is in general non-convex. As such, only local, not global optimality conditions can be guaranteed. Controllers designed in this manner are referred to as within-stride feedback controllers, because the feedback depends exclusively on the continuous-time dynamics that evolve strictly between the instances of consecutive foot-to-floor impacts. In chapter 7, the book also gives a systematic design procedure for event-based feedback controllers, where the parameters of the control law are switched from one set to another at each instance of impact. The utility of the event-based feedback controller is that a continuum of different walking-rates can be achieved. To illustrate these capabilties, both the “within-stride” and the “event-based” feedback controllers have been implemented on actual bipedal robots [1], and these results have been nicely summarized in chapter 8. The remaining chapters present additional interesting results, many of which have practical importance. However, the theoretical content of most of these results seems to be a straightforward adjustment of results attained in previous chapters. For instance, the principles of invariance and attractivity are revisited in chapter 9 to establish stable running, in the same manner that stable walking was established in chapter 6. Finally, chapters 10 and 11 introduce variations of the standard control approach that are tailored for bipedal robots with nontrivial feet and actuated ankles.
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III. COMMENTS Under a very mild set of hybrid system hypotheses, a useful feedback paradigm is introduced which enables the authors to develop strong stability assessment tests. Unlike most results derived from Poincaré’s method, these results are presented in closed-form, thanks in part to the interplay between the feedback and the mechanics. In particular, walking and running gaits are constrained by feedback to lower dimensional invariant manifolds within the robot’s configuration space, so that the problem of limb coordination becomes an optimization problem where the stability requirements can be expressed explicitly as nonlinear inequality constraints.
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Although the approach of the book is mathematically sophisticated, the presentation is made with much clarity and precision. Moreover, all mathematical definitions and peripheral notions needed to follow the text are given a concise, but adequate, explanation in the appendices. As such, the book self-contained. Plenty of illustrative biped robot examples are given—including simulations and experimental data which demonstrate the performance and stability robustness of the proposed class of controllers on real bipedal robot testbeds. For researchers within the controls community, it is worth noting that the results of chapter 4 are not necessarily limited to bipedal robotics. Conceivably, other classes of hybrid systems (e.g., event-driven nonlinear oscillators) could also be investigated using the proposed approach.
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REFERENCES [1] C. Chevallereau, G. Abba, Y. Aoustin, F. Pleastan, E. R. Westervelt, C. Canudas, and J. W. Grizzle, “Rabbit: A testbed for advanced control theory,” IEEE Control Syst. Mag., vol. 23, no. 5, pp. 57–79, Oct. 2003. [2] C. Chevallereau, A. Formal’sky, and B. Perrin, “Low energy cost reference trajectories for a biped robot,” in Proc. IEEE Int. Conf. Robot. Autom., Leuven, Belgium, 1998, pp. 1398–1404. [3] W. M. Haddad, Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control. Princeton, NJ: Princeton Univ. Press, 2006. [4] A. Isidori, Nonlinear Control Systems., 3rd ed. Berlin, Germany: Springer-Verlag, 1995. [5] H. K. Khalil, Nonlinear Systems, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1996.
