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