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The Spring Loaded Inverted Pendulum (SLIP) has been extensively studied and used as a model capturing general aspects of legged locomotion. Biological data suggest that legs regulate energy production and removal via muscle activation, and therefore the conservative SLIP model cannot fully explain the robustness of many legged animals during running and hopping gaits. In this work we consider the active SLIP model: an energetically non-conservative version of the SLIP model with added series actuation. In particular, we propose a partial feedback linearization action for actuator displacement to analytically solve part of its dynamics, thereby reducing computational time and increasing the practicality of performing online control actions. This is then paired with a two-part control action to add/remove energy to/from the system and modify the upcoming apex state to span an open set within the reachable apex states. In addition, we develop two control strategies for online computation of actuator displacement and leg positioning: one to drive the system to a desired state, even in the presence of terrain perturbation; the other to control the system to hop on a desired set of terrain footholds. Furthermore, we propose an adaptive control technique for steady-state locomotion on flat terrain to reduce computation errors by the use of an approximation of the leg-angle dynamics during the stance phase, and we demonstrate the proposed strategy on a more dynamically sophisticated planar hopper model.

One of the key topics in robotics is legged locomotion, taking inspiration from walking and bouncing gaits of bipeds and quadrupeds. A fundamental tool for analyzing running and hopping is the spring loaded inverted pendulum (SLIP), thanks to its simplicity but also effectiveness in modeling bouncing gaits. Being completely passive, the SLIP model does not have the ability to modify its net energy, which, for example, can be a prohibitive obstacle when traveling on a terrain with varying heights. The actuated version of the SLIP model, the active SLIP, considered in this paper, includes a series actuator that allows energy variations via compressing or extending the spring. In this work, we investigate how different actuator motions can affect the system's state, and we propose a control strategy, based on graphical and numerical studies of the reachability space, and updated throughout the stance phase, to drive the system to a desired state. We quantify its performance benefits, particularly in serving as an error-recovering method. The objective of our control strategy is not to replace any leg-placement approach proposed by other works, but rather to be paired with any other leg-placement or path planning method. Its main advantage is the ability to reduce the effects of sensing errors and disturbances happening at landing as well as during the stance phase.

Abstract-The Spring Loaded Inverted Pendulum has been extensively studied and used as an inspiration to the study of legged locomotion. Biological data suggest that legs regulate energy production and removal via muscle activation, and therefore the conservative SLIP model cannot fully explain the robustness of many legged animals during running and hopping gaits. In this work we consider the active SLIP model, an energetically non-conservative version of the SLIP model with series actuation. In particular, we propose a strategy for actuator displacement to add/remove energy from the system, and to analytically solve part of its dynamics. Additionally, we develop a control strategy for online actuator displacement to drive the system to a desired state, even in the presence of terrain perturbation.

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