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Miniaturizing legged robot platforms is challenging due to hardware limitations that constrain the number, power density, and precision of actuators at that size. By leveraging design principles of quasi-passive walking robots at any scale, stable locomotion and steering can be achieved with simple mechanisms and open-loop control. Here, we present the design and control of “Zippy”, the smallest self-contained bipedal walking robot at only 3.6 cm tall. Zippy has rounded feet, a single motor without feedback control, and is capable of turning, skipping, and ascending steps. At its fastest pace, the robot achieves a forward walking speed of 25 cm/s, which is10 leg lengths per second, the fastest biped robot of any size by that metric. This work explores the design and performance of the robot and compares it to similar dynamic walking robots at larger scales. 

Miniaturizing legged robot platforms is challenging due to hardware limitations that constrain the number, power density, and precision of actuators at that size. By leveraging design principles of quasi-passive walking robots at any scale, stable locomotion and steering can be achieved with simple mechanisms and open-loop control. Here, we present the design and control of "Zippy", the smallest self-contained bipedal walking robot at only 3.6 cm tall. Zippy has rounded feet, a single motor without feedback control, and is capable of turning, skipping, and ascending steps. At its fastest pace, the robot achieves a forward walking speed of 25 cm/s, which is10 leg lengths per second, the fastest biped robot of any size by that metric. This work explores the design and performance of the robot and compares it to similar dynamic walking robots at larger scales. 

In order to perform highly dynamic and agile maneuvers, legged robots typically spend time in underactuated domains (e.g. with feet off the ground) where the system has limited command of its acceleration and a constrained amount of time before transitioning to a new domain (e.g. foot touchdown). Meanwhile, these transitions can instantaneously change the system’s state, possibly causing perturbations to be mapped arbitrarily far away from the target trajectory. These properties make it difficult for local feedback controllers to effectively recover from disturbances as the system evolves through underactuated domains and hybrid impact events. To address this, we utilize the fundamental solution matrix that characterizes the evolution of perturbations through a hybrid trajectory and its 2-norm, which represents the worst-case growth of perturbations. In this paper, the worst-case perturbation analysis is used to explicitly reason about the tracking performance of a hybrid trajectory and is incorporated in an iLQR framework to optimize a trajectory while taking into account the closed-loop convergence of the trajectory under an LQR tracking controller. The generated convergent trajectories recover more effectively from perturbations, are more robust to large disturbances, and use less feedback control effort than trajectories generated with traditional methods. 

In this letter we present Hybrid iterative Linear Quadratic Estimation (HiLQE), an optimization based offline state estimation algorithm for hybrid dynamical systems. We utilize the saltation matrix, a first order approximation of the variational update through an event driven hybrid transition, to calculate gradient information through hybrid events in the backward pass of an iterative linear quadratic optimization over state estimates. This enables accurate computation of the value function approximation at each timestep. Additionally, the forward pass in the iterative algorithm is augmented with hybrid dynamics in the rollout. A reference extension method is used to account for varying impact times when comparing states for the feedback gain in noise calculation. The proposed method is demonstrated on an ASLIP hopper system with position measurements. In comparison to the Salted Kalman Filter (SKF), the algorithm presented here achieves a maximum of 63.55% reduction in estimation error magnitude over all state dimensions near impact events. 

Model Predictive Control (MPC) is a popular strategy for controlling robots but is difficult for systems with contact due to the complex nature of hybrid dynamics. To implement MPC for systems with contact, dynamic models are often simplified or contact sequences fixed in time in order to plan trajectories efficiently. In this work, we propose the Hybrid iterative Linear Quadratic Regulator (HiLQR), which extends iLQR to a class of piecewisesmooth hybrid dynamical systems with state jumps. This is accomplished by 1) allowing for changing hybrid modes in the forward pass, 2) using the saltation matrix to update the gradient information in the backwards pass, and 3) using a reference extension to account for mode mismatch. We demonstrate these changes on a variety of hybrid systems and compare the different strategies for computing the gradients. We further show how HiLQR can work in a MPC fashion (HiLQR MPC) by 1) modifying how the cost function is computed when contact modes do not align, 2) utilizing parallelizations when simulating rigid body dynamics, and 3) using efficient analytical derivative computations of the rigid body dynamics. The result is a system that can modify the contact sequence of the reference behavior and plan whole body motions cohesively – which is crucial when dealing with large perturbations. HiLQR MPC is tested on two systems: first, the hybrid cost modification is validated on a simple actuated bouncing ball hybrid system. Then HiLQR MPC is compared against methods that utilize centroidal dynamic assumptions on a quadruped robot (Unitree A1). HiLQR MPC outperforms the centroidal methods in both simulation and hardware tests. 

Robots operating in close proximity to humans rely heavily on human trust to successfully complete their tasks. But what are the real outcomes when this trust is violated? Self-defense law provides a framework for analyzing tangible failure scenarios that can inform the design of robots and their algorithms. Studying self-defense is particularly important for ground robots since they operate within public environments, where they can pose a legitimate threat to the safety of nearby humans. Moreover, even if ground robots can guarantee human safety, the perception of a physical threat is sufficient to justify human self-defense against robots. In this paper, we synthesize works in law, engineering, and social science to present four actionable recommendations for how the robotics community can craft robots to mitigate the likelihood of self-defense situations arising. We establish how current U.S. self-defense law can justify a human protecting themselves against a robot, discuss the current literature on human attitudes toward robots, and analyze methods that have been produced to allow robots to operate close to humans. Finally, we present hypothetical scenarios that underscore how current robot navigation methods can fail to sufficiently consider self-defense concerns and the need for the recommendations to guide improvements in the field. 

Entanglements like vines and branches in natural settings or cords and pipes in human spaces prevent mobile robots from accessing many environments. Legged robots should be effective in these settings, and more so than wheeled or tracked platforms, but naive controllers quickly become entangled and stuck. In this paper we present a method for proprioception aimed specifically at the task of sensing entanglements of a robot’s legs as well as a reaction strategy to disentangle legs during their swing phase as they advance to their next foothold. We demonstrate our proprioception and reaction strategy enables traversal of entanglements of many stiffnesses and geometries succeeding in 14 out of 16 trials in laboratory tests, as well as a natural outdoor environment. 

Hybrid dynamical systems, i.e., systems that have both continuous and discrete states, are ubiquitous in engineering but are difficult to work with due to their discontinuous transitions. For example, a robot leg is able to exert very little control effort, while it is in the air compared to when it is on the ground. When the leg hits the ground, the penetrating velocity instantaneously collapses to zero. These instantaneous changes in dynamics and discontinuities (or jumps) in state make standard smooth tools for planning, estimation, control, and learning difficult for hybrid systems. One of the key tools for accounting for these jumps is called the saltation matrix. The saltation matrix is the sensitivity update when a hybrid jump occurs and has been used in a variety of fields, including robotics, power circuits, and computational neuroscience. This article presents an intuitive derivation of the saltation matrix and discusses what it captures, where it has been used in the past, how it is used for linear and quadratic forms, how it is computed for rigid body systems with unilateral constraints, and some of the structural properties of the saltation matrix in these cases. 

In this paper, we present a method for updating robotic state belief through contact with uncertain surfaces and apply this update to a Kalman filter for more accurate state estimation. Examining how guard surface uncertainty affects the time spent in each mode, we derive a novel guard saltation matrix- which maps perturbations prior to hybrid events to perturbations after - accounting for additional variation in the resulting state. Additionally, we propose the use of parameterized reset functions - capturing how unknown parameters change how states are mapped from one mode to the next - the Jacobian of which accounts for additional uncertainty in the resulting state. The accuracy of these mappings is shown by simulating sampled distributions through uncertain transition events and comparing the resulting covariances. Finally, we integrate these additional terms into the “uncertainty aware Salted Kalman Filter”, uaSKF, and show a peak reduction in average estimation error by 24–60% on a variety of test conditions and systems.