Introducing elasticity in the mechanical design can endow robots with the ability of performing efficient and effective periodic motions. Yet, devising controllers that can take advantage of such elasticity is still an open challenge. This letter tackles an instance of this general problem, by proposing a control architecture for executing goal-oriented and efficient point-to-point periodic motions. This is achieved by (i) producing motor torques that excite intrinsic modal oscillations, and simultaneously (ii) adjusting parallel elasticity so to shape the natural modes. Analytical proofs of convergence are provided for the linear approximation. The performance and efficiency (in the sense of low energy expenditure) of the method in the nonlinear case are assessed with extensive simulations and experiments.

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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.

Energy-efficient gaits in walking robots can be obtained by designing elastic systems that exhibit naturally emerging locomotion patterns. Biological legged locomotion serves as inspiration, as animals use different gaits to move at certain speeds while minimizing energy consumption. To understand the underlying dynamics of biological locomotion, simplified models have been proposed. The most common one, the SLIP (Spring Loaded Inverted Pendulum) model, can explain the effect of the radial elasticity of linear legs and helps to explain locomotion patterns, especially for running behaviors, in different legged systems. Unfortunately, the SLIP model is inappropriate for the study of stability of limit cycles in systems with articulated legs, which are most commonly used in real robots. This paper introduces a novel quadrupedal template model featuring articulated elastic legs, non-constant leg stiffness, and dynamic leg swing. Numerical simulation with a continuation approach is used to discover the gaits emerging from the natural dynamics of the model, without imposing any contact sequence a priori. The stability of those gaits is also characterized, in order to facilitate the exploitation of the natural model dynamics for generating locomotion patterns for quadrupedal robots

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.