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

Abstract

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.

Stability of Fixed Points in an Approximate Solution of the Spring-mass Running Model

“…The same approximation of K as ( 27) was obtained in [27] and [29]. Moreover, in [28] it was shown that the leading order of the expansion of…”

“…The above problem can be easily solved numerically using the shooting method, as was done for example in [12]. Graphical illustrations confirming the validity (27) as an approximation of the solution to the boundary problem (28) are presented in [27] and [29]. In Fig.…”

“…Our approach is similar to the one conducted in [24] where the authors use a different approximation strategy stemming from the principles of mechanics, which, however, obscures the error one makes by necessarily approximating the solutions. In contrast to that, our approach is based on rigorous asymptotic analysis that clearly formulates the region of validity of the model [27,28]. Moreover, as our numerical studies indicate, the fixed point of the Poincaré map undergoes a transcritical bifurcation with increasing energy.…”

“…This has been done in the work of Geyer, Seifarth, and Blickhan [24] which is one of the basis of our subsequent results. Moreover, some approaches to analysis of the current model based on approximate solutions have been conducted in [25,26,22] and in our earlier works [27,28]. There, we have conducted a rigorous Poicaré-Lindstedt asymptotic analysis based on the perturbation theory in the regime of large spring stiffness and small angle of attack.…”

Stability of Fixed Points in an Approximate Solution of the Spring-Mass Running Model

“…The above problem can be easily solved numerically using the shooting method, as was done for example in [12]. Graphical illustrations confirming the validity (27) as an approximation of the solution to the boundary problem (28) are presented in [27] and [29].…”