The spring-loaded inverted pendulum is a widely used model of legged locomotion. However, a complete map of the dimensionless parameter regions that correspond to the stable periodic solutions cannot be found in the literature. In this work, the three-dimensional space of two dimensionless physical parameters and the dimensionless total mechanical energy of the conservative system was discovered by means of numerical continuation. The fundament of the stability analysis of the piecewise-smooth system was provided by the numerical calculation of the fundamental solution matrices and the monodromy matrix. An effective iteration procedure based on the Nelder-Mead method is presented which tunes the model parameters in order to imitate the motion characteristics of specific animals and locomotion types such as running, trotting and galloping. The results are available online in the form of an interactive platform.