Application of Koopman Operator Theory to Legged Locomotion

Abstract

Nonlinearities from complicated robot systems and harsh contact dynamics have long impeded the effectiveness of optimal control strategies for legged robots. In this work, we present a linearized simple walking model using Koopman Operator Theory, and its usage in Linear Model Predictive Control (L-MPC). Various walking and contact models were evaluated, but ultimately the rimless wheel was selected due to its inherent stability and low dimensionality, and a nonlinear viscoelastic model was used to accurately capture floor contact and impact dynamics. Koopman models were developed using both Radial Basis Functions (RBFs) and neural network-generated observables for the passive rimless wheel. A novel actuation method with linear actuators, combined with the Control Coherent Koopman methodology, resulted in accurate linear models that effectively enabled L-MPC to control the wheel on flat ground. This model outperformed those created using the more traditional Dynamic Mode Decomposition with Control method. This work demonstrates the power of Koopman linearization to produce a unified set of linear dynamical equations that encompass various contact and non-contact configurations and demonstrates the effectiveness of the Control Coherent Koopman methodology in generating an accurate input matrix across these different contact modes.