Numerical bifurcation analysis of the bipedal spring-mass model

• The spring–mass model is transformed into a two-point boundary value problem (BVP).
• Calculation of stable solutions is reduced to the calculation of their boundaries.
• The complete solution manifold of the model is computed for the first time.
• Our approach can be extended for investigation of gait transitions and variability.
• Our approach is numerically stable. All BVPs can be solved using single shooting.

The spring–mass model and its numerous extensions are currently one of the best candidates for templates of human and animal locomotion. However, with increasing complexity, their applications can become very time-consuming. In this paper, we present an approach that is based on the calculation of bifurcations in the bipedal spring–mass model for walking. Since the bifurcations limit the region of stable walking, locomotion can be studied by computing the corresponding boundaries. Originally, the model was implemented as a hybrid dynamical system. Our new approach consists of the transformation of the series of initial value problems on different intervals into a single boundary value problem. Using this technique, discontinuities can be avoided and sophisticated numerical methods for studying parametrized nonlinear boundary value problems can be applied. Thus, appropriate extended systems are used to compute transcritical and period-doubling bifurcation points as well as turning points. We show that the resulting boundary value problems can be solved by the simple shooting method with sufficient accuracy, making the application of the more extensive multiple shooting superfluous. The proposed approach is fast, robust to numerical perturbations and allows determining complete manifolds of periodic solutions of the original problem.