We present a new open-source torque-controlled legged robot system, with a low cost and low complexity actuator module at its core. It consists of a low-weight high torque brushless DC motor and a low gear ratio transmission suitable for impedance and force control. We also present a novel foot contact sensor suitable for legged locomotion with hard impacts. A 2.2 kg quadruped robot with a large range of motion is assembled from 8 identical actuator modules and 4 lower legs with foot contact sensors. To the best of our knowledge, it is the most lightest force-controlled quadruped robot. We leverage standard plastic 3D printing and off-theshelf parts, resulting in light-weight and inexpensive robots, allowing for rapid distribution and duplication within the research community. In order to quantify the capabilities of our design, we systematically measure the achieved impedance at the foot in static and dynamic scenarios. We measured up to 10.8 dimensionless leg stiffness without active damping, which is comparable to the leg stiffness of a running human. Finally, in order to demonstrate the capabilities of our quadruped robot, we propose a novel controller which combines feedforward contact forces computed from a kino-dynamic optimizer with impedance control of the robot center of mass and base orientation. The controller is capable of regulating complex motions which are robust to environmental uncertainty. *

Abstract-This work explores the possible roles of active tails for steady-state legged-locomotion. A series of simple models are proposed which capture the dynamics of an idealized running system with an active tail. The models suggest that the control objectives of injecting energy into the system and stabilizing body-pitch can be effectively decoupled via proper tail design: a long, light tail. Thus the overall control problem can be simplified, using the tail exclusively to stabilize body-pitch: this effectively relaxes the constraints on the leg-actuators, allowing them to be recruited specifically for adding energy into the system. We show in simulation that models with long-light tails are better able to reject perturbations to body-pitch than short-heavy tails with the same moment of inertia. Further, we present the results of a one degree-of-freedom tail mounted on the open-loop controlled quadruped robot Cheetah-Cub. Our results show that an active tail can greatly improve both forward velocity and reduce body-pitch per stride, while adding minimal complexity. Further, the results validate the long-light tail design: shorter, heavier tails are much more sensitive to configuration and control parameter changes than longer and lighter tails with the same moment of inertia.

It is currently unclear if damping plays a functional role in legged locomotion, and simple models often do not include damping terms. We present a new model with a damping term that is isolated from other parameters: that is, the damping term can be adjusted without retuning other model parameters for nominal motion. We systematically compare how increased damping affects stability in the face of unexpected ground-height perturbations. Unlike most studies, we focus on task-level stability: instead of observing whether trajectories converge towards a nominal limit-cycle, we quantify the ability to avoid falls using a recently developed mathematical measure. This measure allows trajectories to be compared quantitatively instead of only being separated into a binary classification of ‘stable' or ‘unstable'. Our simulation study shows that increased damping contributes significantly to task-level stability; however, this benefit quickly plateaus after only a small amount of damping. These results suggest that the low intrinsic damping values observed experimentally may have stability benefits and are not simply minimized for energetic reasons. All Python code and data needed to generate our results are available open source.

Properly designing a system to exhibit favorable natural dynamics can greatly simplify designing or learning the control policy. However, it is still unclear what constitutes favorable natural dynamics and how to quantify its effect. Most studies of simple walking and running models have focused on the basins of attraction of passive limit-cycles and the notion of self-stability. We instead emphasize the importance of stepping beyond basins of attraction. We show an approach based on viability theory to quantify robust sets in state-action space. These sets are valid for the family of all robust control policies, which allows us to quantify the robustness inherent to the natural dynamics before designing the control policy or specifying a control objective. We illustrate our formulation using spring-mass models, simple low dimensional models of running systems. We then show an example application by optimizing robustness of a simulated planar monoped, using a gradient-free optimization scheme. Both case studies result in a nonlinear effective stiffness providing more robustness.