Stable walking on variable visco-elastic terrains using meta-parameters for passive state migration

This paper investigates how a walker could estimate the variability of an arbitrary set of state variables when migrating on visco-elastic grounds. The state variables are a function of both the visco-elastic settings of the walking body and soft terrain parameters. A rimless wheel model was developed using a Lagrangian approach in order to obtain analytical solutions for migration across ground conditions. An algorithm was then developed to determine the steady value of the variables as a function of the difference in ground and hub parameters involved in the migration. A generalised migration metaparameter, Δ_g, function of this difference, was then extrapolated using polynomial approximation. Δ_g can be used to estimate the expected variability at a state given information on actual and previous ground parameters. A second parameter, Δ_h, describing local variability of a given state on a given terrain, is used to generate a predictive algorithm capable of stabilising the rimless wheel setup when subject to an abrupt change in ground parameters. We actuate the rimless wheel with a constant torque leaving it to develop any speed profile for a given visco-elastic impedance distribution of the ground and its own vertical visco-elastic impedance. The ground is altered depending on the two migration meta-parameters (Δ_g and Δ_r), ensuring both local and migration stability.