  Spring Loaded Inverted Pendulum (SLIP) model has a long history in describing
running behavior in animals and humans as well as has been used as a design
basis for robots capable of dynamic locomotion. Anchoring the SLIP for lossy
physical systems resulted in newer models which are extended versions of
original SLIP with viscous damping in the leg. However, such lossy models
require an additional mechanism for pumping energy to the system to control the
locomotion and to reach a limit-cycle. Some studies solved this problem by
adding an actively controllable torque actuation at the hip joint and this
actuation has been successively used in many robotic platforms, such as the
popular RHex robot. However, hip torque actuation produces forces on the COM
dominantly at forward direction with respect to ground, making height control
challenging especially at slow speeds. The situation becomes more severe when
the horizontal speed of the robot reaches zero, i.e. steady hoping without
moving in horizontal direction, and the system reaches to singularity in which
vertical degrees of freedom is completely lost. To this end, we propose an
extension 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 vertical
direction. We propose an approximate analytical solution to the nonlinear
system dynamics of SLIP- SCM model to characterize its behavior during the
locomotion. Finally, we perform a fixed-point stability analysis on SLIP-SCM
model using our approximate analytical solution and show that proposed model
exhibits stable behavior in our range of interest.
