Planar hopping control strategy for tail-actuated SLIP model traversing varied terrains

Biologically inspired by the hopping performance of kangaroo, this paper extends the traditional Spring-loaded Inverted Pendulum (SLIP) model by adding an actuation at hip to composite a tail-actuated SLIP (TSLIP) model as an abstract template for gait controller design. To deal with the intrinsic nonlinearity of stance dynamics in the TSLIP model, an analytical approximation is derived by virtue of perturbation technique with gravity correction. Employing the derived solution to construct the apex return map, a gait controller is further devised with a two-layer nonlinear optimization scheme. The outer loop optimizes the tail motion during stance by matching the energy variation between the current and target apex state while the inner loop subsequently selects the optimum touchdown angle by minimizing the difference between the predictive and target apex vector from stride to stride. Additionally, an extended control strategy that embodies a time-dependent pre-positioned policy for swing-leg working in conjunction with the active tail is devised, requiring no priori knowledge of the ground truth to enhance to hopping performance of the TSLIP system. The simulation results have demonstrated the effectiveness of the proposed control strategy for tailed hopping system.