We present the supersymmetric version of the minimal Randall-Sundrum model with two opposite tension branes.

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

RHex is an untethered, compliant leg hexapod robot that travels at better than one body length per second over terrain few other robots can negotiate at all. Inspired by biomechanics insights into arthropod locomotion, RHex uses a clock excited alternating tripod gait to walk and run in a highly maneuverable and robust manner. We present empirical data establishing that RHex exhibits a dynamical ("bouncing") gait-its mass center moves in a manner * This work was supported in part by DARPA/SPAWAR under contract N66001-00-C-8026. Portions of the material reported here were first presented in a conference paper appearing in the collection (Altendorfer et al., 2000). 208 Altendorfer et al. well approximated by trajectories from a Spring Loaded Inverted Pendulum (SLIP)-characteristic of a large and diverse group of running animals, when its central clock, body mass, and leg stiffnesses are appropriately tuned. The SLIP template can function as a useful control guide in developing more complex autonomous locomotion behaviors such as registration via visual servoing, local exploration via visual odometry, obstacle avoidance, and, eventually, global mapping and localization.

We present a new stability analysis for hybrid legged locomotion systems based on the "symmetric" factorization of return maps.We apply this analysis to two-degrees-of-freedom (2DoF) and threedegrees-offreedom (3DoF) models of the spring loaded inverted pendulum (SLIP) with different leg recirculation strategies. Despite the non-integrability of the SLIP dynamics, we obtain a necessary condition for asymptotic stability (and a sufficient condition for instability) at a fixed point, formulated as an exact algebraic expression in the physical parameters. We use this expression to characterize analytically the sensory cost and stabilizing benefit of various feedback schemes previously proposed for the 2DoF SLIP model, posited as a lowdimensional representation of running.We apply the result as well to a 3DoF SLIP model that will be treated at greater length in a companion paper as a descriptive model for the robot RHex.

We apply the stability analysis for hybrid legged locomotion systems, introduced in our companion paper in this issue, to a new simple clock-driven SLIP model inspired by the robot RHex. We adopt in stance phase the three-degrees-of-freedom (3DoF) spring loaded inverted pendulum (SLIP) model introduced in our companion paper to capture RHex’s pitching dynamics in the sagittal plane. The coordinating influence of RHex’s open-loop clock controller is subsumed into a leg placement strategy derived from a bipedal abstraction of RHex. The “symmetric” factorization analysis introduced in our companion paper yields a necessary condition for gait stability expressed in closed form, which can be imposed directly on the clock parameter space. This represents the first reported analytical insight into how a dynamical runner might be stabilized by a completely feed forward rhythmic limb coordination pattern. Correspondence in steady-state gait location and stability characteristics with an appropriately tuned 24DoF model of RHex provides numerical evidence that the 3DoF SLIP model offers a descriptive explanation for the robot’s empirical running behavior.