Eigenmodes of Nonlinear Dynamics: Definition, Existence, and Embodiment into Legged Robots With Elastic Elements

Pogo-stick bouncing or the spring loaded inverted pendulum represents fundamental dynamics models for hopping and running in legged locomotion. However, these conceptual models are in general of lower order than the elastic multibody dynamics of versatile segmented legs. The question how to embody these simple models into real robot leg designs still has not been completely answered so far. The concept of eigenmodes for linear systems provides a tool to separate high-dimensional, coupled dynamics in one-dimensional (1-D) invariant ones. However, the dynamics of segmented legs is in general nonlinear such that even the existence of periodic motions, as appearing typically in locomotion tasks, cannot be generally guaranteed without changing intrinsic dynamics behavior substantially by control. This letter extends the concept of eigenmodes, which is well known for linear systems, to the nonlinear case. By proposing a method for selecting the design parameters of multibody systems such that desired eigenmodes are achieved, the problem of embodying fundamental locomotion modes into legged systems is resolved. Examples of practically realizable leg designs are provided, which proof the existence of invariant, 1-D oscillation modes in nonlinear, elastic robot dynamics. An experiment on a multilegged robotic system validates that energetic efficiency can be gained by the proposed approach.