A Simply Stabilized Running Model

Abstract

Abstract. The spring-loaded inverted pendulum (SLIP), or monopedal hopper, is an archetypal model for running in numerous animal species. Although locomotion is generally considered a complex task requiring sophisticated control strategies to account for coordination and stability, we show that stable gaits can be found in the SLIP with both linear and "air" springs, controlled by a simple fixed-leg reset policy. We first derive touchdown-to-touchdown Poincaré maps under the common assumption of negligible gravitational effects during the stance phase. We subsequently include and assess these effects and briefly consider coupling to pitching motions. We investigate the domains of attraction of symmetric periodic gaits and bifurcations from the branches of stable gaits in terms of nondimensional parameters.Key words. legged locomotion, spring-loaded inverted pendulum, periodic gaits, bifurcation, stability AMS subject classifications. 34C23, 37J20, 37J25, 37J60, 70Hxx, 70K42, 70K50PII. S11111111024083111. Introduction. Locomotion, "moving the body's locus," is among the most fundamental of animal behaviors. A large motor science literature addresses gait pattern selection [1], energy expenditure [2], underlying neurophysiology [3], and coordination in animals and machines [4]. In this paper, we explore the stabilizing effect of a very simple control policy on a very simple running model.Legged locomotion is generally considered a complex task [5] involving the coordination of many limbs and redundant degrees of freedom [6]. In [7], Full and Koditschek note that "locomotion results from complex, high-dimensional, non-linear, dynamically coupled interactions between an organism and its environment." They distinguish locomotion models simplified for the purpose of task specification (templates) from more kinematically and dynamically accurate representations of the true body morphology (anchors). A template is a formal reductive model that (1) encodes parsimoniously the dynamics of the body and its payload transport capability, using the minimum number of variables and parameters, and (2) advances an intrinsic hypothesis concerning the control strategy underlying the achievement of this task. Anchors are not only more elaborate dynamical systems grounded in the morphology and physiology

Spring-mass running: simple approximate solution and application to gait stability

“…At this point, one either proceeds by resuming to numerical studies (e.g. Ghigliazza et al, 2003), or further simplifications are introduced. For instance, in Schwind and Koditschek (2000) the mean value theorem is applied to circumvent elliptic integrals yielding a good approximation of the modified system dynamics in stance, especially, when the spring compression is close to its maximum.…”

“…in Schwind and Koditschek (2000) mean errors as high as 20% for the radial velocity at take-off are observed). Although these solutions may be used for qualitative assessments (Ghigliazza et al, 2003), any quantitative result seems highly questionable. Bearing in mind that the spring-mass model is employed to devise general control schemes for running machines (Raibert, 1986;Saranli and Koditschek, 2003) and to investigate animal and human locomotion (He et al, 1991;Farley et al, 1991Farley et al, , 1993Seyfarth et al, 2001), this leaves a rather unsatisfactory state.…”

“…Although the dynamics of the central force system could have been obtained by consequently solving elliptic integrals, this approach avoids the difficult quadratures that typically remain even when gravity is ignored (e.g. Schmitt and Holmes, 2000;Ghigliazza et al, 2003). Hence, the radial and angular motions can be extracted in terms of elementary functions.…”

“…For instance, by mapping the model's parameter space, simulation studies suggest that the spring-mass system for running can display a 'self-stable' behavior Ghigliazza et al, 2003). Here, selfstability refers to the observation of asymptotically stable gait trajectories without continuous sensory feedback.…”

“…Ghigliazza et al, 2003). As this can hardly be done in general locomotion (Schwind and Koditschek, 2000) or when addressing physiologically motivated parameters (Geyer, 2001), in Schwind and Koditschek (2000) an iterative algorithm reincorporating the effect of gravity is introduced.…”