To analyse walking, running or hopping motions, models with high degrees of freedom are usually used. However simple reductionist models are advantageous within certain limits. In a simple manner, the hopping motion is generally modelled by a spring-mass system, resulting in piecewise smooth dynamics with marginally stable periodic solutions. For a more realistic behaviour, the spring is replaced by a variety of muscle models due to which asymptotically stable periodic motions may occur. The intrinsic properties of the muscle model, i.e. preflexes, are usually taken into account in three complexities—constant, linear and Hill-type. In this paper, we propose a semi-closed form feed-forward control which represents the muscle activation and results in symmetrical hopping motion. The research question is whether hopping motions with symmetric force-time history have advantages over asymmetric ones in two aspects. The first aspect is its applicability for describing human motion. The second aspect is related to robotics where the efficiency is expressed in term of performance measures. The symmetric systems are compared with each other and with those from the literature using performance measures such as hopping height, energetic efficiency, stability of the periodic orbit, and dynamical robustness estimated by the local integrity measure (LIM). The paper also demonstrates that the DynIn MatLab Toolbox that has been developed for the estimation of the LIM of equilibrium points is applicable for periodic orbits.

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.