Several template models have been developed to facilitate the analysis of limit-cycles for quadrupedal locomotion. The parameters in the model are usually fixed; however, biology shows that animals change their leg stiffness according to the locomotion velocity, and this adaptability invariably affects the stability of the gait. This paper provides an analysis of the influence of this variable leg stiffness on the stability of different quadrupedal gaits. The analysis exploits a simplified quadrupedal model with compliant legs and shoulder joints represented as torsional springs. This model can reproduce the most common quadrupedal gaits observed in nature. The stability of such emerging gaits is then checked. Afterward, an optimization process is used to search for the system parameters that guarantee maximum gait stability. Our study shows that using the highest feasible leg swing frequency and adopting a leg stiffness that increases with the speed of locomotion noticeably improves the gait stability over a wide range of horizontal velocities while reducing the oscillations of the trunk. This insight can be applied in the design of novel elastic quadrupedal robots, where variable stiffness actuators could be employed to improve the overall locomotion behavior.

Simple template models have proven useful for understanding the underlying dynamics of legged locomotion. The most common one, the SLIP model, considers the legs as linear springs with constant stiffness, and it explains well the radial dynamics of the legs. However, in order to study the influence of the leg swing dynamics and leg segmentation on gait stability, more complex models are required. This paper introduces a novel template model for quadrupedal gait, which considers these additional aspects. The dynamic behavior of the model is analyzed via numerical simulation, using a continuation approach. By conducting a parametric analysis on the trotting gait and analyzing its stability, we identify the influence of the main model parameters, leading to marginally unstable limit cycles. These numerical results are applicable to the design of more efficient elastic quadrupedal robots.

Robotic legs often lag behind the performance of their biological counterparts. The inherent passive dynamics of natural legs largely influences the locomotion and can be abstracted through the spring-loaded inverted pendulum (SLIP) model. This model is often approximated in physical robotic legs using a leg with minimal mass. Our work aims to embed the SLIP dynamics by using a nonlinear strict oscillatory mode into a segmented robotic leg with significant mass, to minimize the control required for achieving periodic motions. For the first time, we provide a realization of a nonlinear oscillatory mode in a robotic leg prototype. This is achieved by decoupling the polar task dynamics and fulfilling the resulting conditions with the physical leg design. Extensive experiments validate that the robotic leg effectively embodies the strict mode. The decoupled leg-length dynamic is exhibited in leg configurations corresponding to the stance and flight phases of the locomotion task, both for the passive system and when actuating the motors. We additionally show that the leg retains this behavior while performing jumping in place experiments.