Spring Loaded Inverted Pendulum (SLIP) model has a long history in describingrunning behavior in animals and humans as well as has been used as a designbasis for robots capable of dynamic locomotion. Anchoring the SLIP for lossyphysical systems resulted in newer models which are extended versions oforiginal SLIP with viscous damping in the leg. However, such lossy modelsrequire an additional mechanism for pumping energy to the system to control thelocomotion and to reach a limit-cycle. Some studies solved this problem byadding an actively controllable torque actuation at the hip joint and thisactuation has been successively used in many robotic platforms, such as thepopular RHex robot. However, hip torque actuation produces forces on the COMdominantly at forward direction with respect to ground, making height controlchallenging especially at slow speeds. The situation becomes more severe whenthe horizontal speed of the robot reaches zero, i.e. steady hoping withoutmoving in horizontal direction, and the system reaches to singularity in whichvertical degrees of freedom is completely lost. To this end, we propose anextension of the lossy SLIP model with a slider-crank mechanism, SLIP- SCM,that can generate a stable limit-cycle when the body is constrained to verticaldirection. We propose an approximate analytical solution to the nonlinearsystem dynamics of SLIP- SCM model to characterize its behavior during thelocomotion. Finally, we perform a fixed-point stability analysis on SLIP-SCMmodel using our approximate analytical solution and show that proposed modelexhibits stable behavior in our range of interest.
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