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In this paper we present a closed-loop optimal control approach for the online control of a legged robot locomotion, particularly the hopping of a simulated monoped robot. Modeling is done based on the spring loaded inverted pendulum (SLIP) model suggested as the animal and human running gait template. The key idea is to efficiently inject energy to the system so that the monoped can track the desired apex height and forward velocity. The state of the system is observed in the Poincaré section at the apex point and the corresponding discrete dynamics is formulated by using available analytical solutions. The goal is then to synthesize an optimal control law which can bring the apex state at any step to the desired state at the next step. We show the controller performance in providing fast and accurate response in the presence of noise and through different scenarios while minimizing the control effort.

This work explores the use of active tails for steady-state legged-locomotion. Simple models are proposed which capture the dynamics of an idealized running system with an active tail. Analysis suggests that the control objectives of injecting energy into the system and stabilizing body-pitch can be effectively decoupled via proper tail design: a long, light tail. Thus the overall control problem can be simplified, using the tail exclusively to stabilize body-pitch. We show in simulation that models with long-light tails are better able to reject perturbations to body-pitching than short-heavy tails with the same moment of inertia. We also present the results of an active tail mounted on the quadruped robot Cheetah-Cub. The results show greatly improved forward velocity and reduced body-pitching and validate the long-light tail design: shorter, heavier tails are much more sensitive to control parameter changes.