Predicting bifurcation of mechanical systems using reservoir computing: Case studies on legged locomotion and pneumatic soft actuator

Understanding and predicting how mechanical systems respond to environmental variability is essential for advancing next-generation robotic systems with physical intelligence. In this study, we investigated the use of echo state networks (ESNs), a representative class of reservoir computing (RC) models, to predict the bifurcation structures of real-world mechanical systems from limited observations. We examined two representative cases: a simulated passive dynamic walking (PDW) robot with hybrid continuous-discrete dynamics and a real-world soft pneumatic artificial muscle (PAM) actuator whose electrical resistance undergoes complex changes under varying loads. To address the challenges posed by the PDW's hybrid dynamics, we proposed a hybrid ESN (HESN) model that integrates a knowledge-based touchdown detection mechanism with an ESN module. The HESN successfully reproduced the route-to-chaos bifurcation structure of the PDW, captured its multi-attractor dynamics, and demonstrated robustness against imperfect domain knowledge. For the PAM, where no reliable physical model is available, a purely data-driven ESN accurately predicted resistance bifurcations across changing environmental conditions. These results highlight the potential of RC models as flexible digital twins for mechanical systems, enabling parameter-aware modeling of bifurcations with limited training data and supporting the design of robust, adaptive robots capable of operating in complex environments.