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Mechanical Aspects of Legged Locomotion Control Arthropod Structure & Development 33 (2004) 251–272 www.elsevier.com/locate/asd Mechanical aspects of legged locomotion control Daniel E. Koditscheka,*, Robert J. Fullb,1, Martin Buehlerc,2 aAI Lab and Controls Lab, Department of EECS, University of Michigan, 170 ATL, 1101 Beal Ave., Ann Arbor, MI 48109-2110, USA bPolyPEDAL Laboratory, Department of Integrative Biology, University of California at Berkeley, Berkeley, CA 94720-3140, USA cRobotics, Boston Dynamics, 515 Massachusetts Avenue, Cambridge, MA 02139, USA Received 9 March 2004; accepted 28 May 2004 Abstract We review the mechanical components of an approach to motion science that enlists recent progress in neurophysiology, biomechanics, control systems engineering, and non-linear dynamical systems to explore the integration of muscular, skeletal, and neural mechanics that creates effective locomotor behavior. We use rapid arthropod terrestrial locomotion as the model system because of the wealth of experimental data available. With this foundation, we list a set of hypotheses for the control of movement, outline their mathematical underpinning and show how they have inspired the design of the hexapedal robot, RHex. q 2004 Elsevier Ltd. All rights reserved. Keywords: Insect locomotion; Hexapod robot; Dynamical locomotion; Stable running; Neuromechanics; Bioinspired robots 1. Introduction: an integrative view of motion science challenge is to discover the secrets of how they function collectively as an integrated whole. These systems possess Motion science has not yet been established as a single functional properties that emerge only upon interaction with clearly definable discipline, since the relevant knowledge one another and the environment. Our goal is to uncover the base spans the range of biology (Alexander, 2003; control architectures that result in rapid arthropod runners Biewener, 2003; Daniel and Tu, 1999; Dickinson et al., being remarkably stable and possessing the same pattern of 2000; Full, 1997; Grillner et al., 2000; Pearson, 1993), whole body mechanics as reptiles, birds and mammals medicine (Winters and Crago, 2000), psychology (Haken (Blickhan and Full, 1993). Guided by experimental et al., 1985), mathematics (Guckenheimer and Holmes, measurements, mathematical models and physical (robot) 1983) and engineering (Ayers et al., 2002). Locating the models, we postulate control architectures that necessarily origin of control remains a substantial research challenge, include the constraints of the body’s mechanics. We exploit the because neural and mechanical systems are dynamically fact that body and limbs must obey inertia-dominated New- coupled to one another, and both play essential roles in tonian mechanics to constrain possible control architectures. control. While it is possible to deconstruct the mechanics of This paper reviews the locomotion control hierarchy as a locomotion into a simple cascade—brain activates muscles, series of biologically inspired hypotheses that have given muscles move skeleton, skeleton performs work on external rise to a novel robot and that we are just beginning to world—such a unidirectional framework fails to incorporate translate into specific biologically refutable propositions. essential complex dynamic properties that emerge from Here, we focus on the lowest end of this neuromechanical feedback operating between and within levels. The major hierarchy where we hypothesize the primacy of mechanical feedback or ‘preflexes’—neural clock excited tuned * Corresponding author. Tel.: þ1-734-764-4307; fax: þ1-734-763-1260. muscles acting through chosen skeletal postures (Brown E-mail addresses: [email protected] (D.E. Koditschek), http://ai.eecs. and Loeb, 2000). Such notions are most succinctly umich.edu/people/kod, [email protected] (R.J. Full), http:// expressed in the mathematical language of mechanics and polypedal.berkeley.edu/., [email protected] (M. Buehler), dynamical systems theory. We view this paper, on one level, http://www.bostondynamics.com.. 1 Tel.: þ1-510-643-5183; fax: þ1-510-643-6264. as a guide for the interested reader to the narrower technical 2 Tel.: þ1-617-868-5600x235. literature within which these ideas have found their clearest 1467-8039/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.asd.2004.06.003 252 D.E. Koditschek et al. / Arthropod Structure & Development 33 (2004) 251–272 (albeit incomplete, since the underlying mathematics is still summarily in Fig. 10, representing diagrammatically a plane far from worked out) expression. However, we intend as of alternatives spanning on the one hand a range between well that this presentation should be sufficiently explanatory pure feedback and pure feedforward control options, and, on as to stand alone for those outside the engineering and the other, a range between completely centralized and applied mathematics community, as an account of what we completely decentralized computational options. We presently do and do not understand about the locomotion hypothesize a relationship between the ‘noise’ in the control hierarchy associated with the new machine, RHex. internal communication paths or internal computational Motivated by the view that synthesis is the final arbiter of world model, the time constants demanded by the physical understanding, we present the procession of inspiration, task, and the preferred operating point to support its insight and implementation flowing from the biology execution on this architectural plane. Once again, this toward engineering RHex, the most agile, autonomous predicted set of relationships is explored using the legged robot yet built (Altendorfer et al., 2001; Buehler illustrative example of the physical model, RHex, and the et al., 2002; Saranli et al., 2001), and the growing very recent empirical relationships we have begun to mathematical insight into locomotion arising in conse- observe between style of control or communication scheme quence. We organize the presentation of this flow from and efficacy of behavior. biology to engineering into three distinct conceptual areas A brief conclusion reviews the nature of these hypoth- as follows: (i) how the traditions of dynamical systems eses, and closes with the necessary humbling comparison of theory inform the overall framework and provide a point of RHex to the wonderful, far more impressive locomotion departure for our work, addressed by hypothesis H1; (ii) how capabilities of animals whose performance still far exceeds a purely mechanical view of body morphology and what we yet understand about, and even farther exceeds materials design in conjunction with those dynamical what we know how to build into legged running systems. systems theoretic ideas can begin to explain important features of animal locomotion, addressed by hypothesis H2; and (iii) how the crucial and voluminous pre-existing data 2. Stability-dynamical systems approach to motion (arising from decades of painstaking work in neuroethol- science ogy) about animals can offer hints on the manner in which a nervous system might be effectively coupled to the type of Stability is essential to the performance of terrestrial tunable mechanical system just described, addressed by locomotion. Arthropods are often viewed as the quintessen- hypothesis H3. tial example of a statically stable design. Arthropod legs Hypothesis H1, introduced in Section 2, comprises a generally radiate outwards, providing a wide base of general orientation to the ideas of dynamical systems theory support. Their center of mass is often so low that their and their applicability to dynamical running. It asserts that body nearly scrapes the ground. Their sprawled postures the primary requirement of an animal’s locomotion strategy reduce over-turning moments. Hughes (1952) argued that is to stabilize its body around steady state periodic motions the six legged condition is the ‘end-product of evolution’ termed limit cycles. The section is concerned with elaborating because the animal can always be statically stable—at least the implications of this view as focused on patterns of three legs are planted on the ground with the center of mass mechanical response to perturbation, and reviewing the within the triangle of support. longstanding role dynamical systems thinking has had in the development of agile robot runners, including RHex. 2.1. Dynamic stability in arthropod running We next introduce in Section 3 hypothesis H2 proposing a specific solution to Bernstein’s famous ‘degrees of Statically stable design for slower arthropod locomotion freedom’ problem (Bernstein, 1967), representing a purely does not preclude dynamic effects at faster speeds (Ting mechanical explanation for the appearance of synergies in et al., 1994). Results from the study of six and eight-legged animal locomotion. It posits the representation of a motor runners (Blickhan and Full, 1987; Full and Tu, 1990, 1991; task via a low degree of freedom template dynamical system Full et al., 1991) provide strong evidence that dynamic that is anchored via the selection of a preferred posture. The stability cannot be ignored in fast, multi-legged runners that section underscores the intrinsic role that dynamical are maneuverable. In running cockroaches, several loco- systems thinking plays in the development of this motor metrics change in a direction that is consistent with an hypothesis, and explores some of its specific empirical increase in the importance of dynamic stability as speed concomitants through the illustrative example
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