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; 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 nonconservative 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, and the other to control the system to hop on a desired set of terrain footholds. Furthermore, we demonstrate the proposed strategy on a more dynamically sophisticated planar hopper model.

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.

The Spring-Loaded Inverted Pendulum (SLIP) is considered the simplest model to effectively describe bouncing gaits (such as running and hopping) for many legged animals and robots. For this reason, it is has often been used as a model for robot design. A key challenge in using this model, however, is the lack of a closed-form solution for the equations of motion that define the stance phase of its dynamics. This results in the impossibility of analytically predicting its trajectory. Consequently, developing a practical control strategy to operate on the model is computationally intensive, because accurately predicting the step-to-step dynamics is still an unsolved problem. By adding an actuator in series with the spring, we can develop a control law for actuator displacement which enforces a desired trajectory during stance. In particular, for our specific chosen control law, we can compute an analytical solution for the stance phase trajectory. Furthermore, we give examples of higher level control strategies for foothold placement and for keeping the forward velocity or the apex height constant on rough terrain that employ our low-level control laws, and we illustrate through simulations the performance typical of our strategy.

This paper focuses on Euler angles and on the decomposition of rotations. We consider arbitrary rotation axes that are not necessarily mutually orthogonal; we characterize the set of rotation matrices that admit Euler angles about arbitrary rotation axes; and we provide a single set of Euler angle formulas that applies to any selection of rotation axes. The results are presented and derived in a coordinate-free setting, where no reference frames are required, and no components of any array or matrix are manipulated.