  We consider the classic spring-mass model of running which is built upon an
inverted elastic pendulum. In a natural way, there arises an interesting
boundary value problem for the governing system of two nonlinear ordinary
differential equations. It requires us to choose the stiffness to ascertain
that after a complete step, the spring returns to its equilibrium position.
Motivated by numerical calculations and real data we conduct a rigorous
asymptotic analysis in terms of the Poicar\'e-Lindstedt series. The
perturbation expansion is furnished by an interplay of two time scales what has
an significant impact on the order of convergence. Further, we use these
asymptotic estimates to prove that there exists a unique solution to the
aforementioned boundary value problem and provide an approximation to the
sought stiffness. Our results rigorously explain several observations made by
other researchers concerning the dependence of stiffness on the initial angle
of the stride and its velocity. The theory is illustrated with a number of
numerical calculations.
