Asymptotic Solution of a Boundary Value Problem for a Spring–Mass Model of Legged Locomotion

Running is the basic mode of fast locomotion for legged animals. One of the most successful mathematical descriptions of this gait is the so-called spring–mass model constructed upon an inverted elastic pendulum. In the description of the grounded phase of the step, an interesting boundary value problem arises where one has to determine the leg stiffness. In this paper, we find asymptotic expansions of the stiffness. These are conducted perturbatively: once with respect to small angles of attack, and once for large velocities. Our findings are in agreement with previous results and numerical simulations. In particular, we show that the leg stiffness is inversely proportional to the square of the attack angle for its small values, and proportional to the velocity for large speeds. We give exact asymptotic formulas to several orders and conclude the paper with a numerical verification.

Dynamical behaviour of a compound elastic pendulum

The aim of the paper is a comprehensive study of the compound elastic pendulum (CEP) with two degrees of freedom to point out the main complex (chaotic) dynamics that it can exhibit. The simplest way to find complex behavior in a nonintegrable Hamiltonian system is firstly to look for homoclinic (heteroclinic) orbit(s). Here, under suitable assumptions, we detect the existence of a homoclinic orbit of CEP and present the equation for it. Moreover, we show that for any value of the small parameter the system has a hyperbolic periodic orbit, whose invariant manifolds intersect themselves transversally.