This letter presents a new notion of input-tostate safe control barrier functions (ISSf-CBFs), which ensure safety of nonlinear dynamical systems under input disturbances. Similar to how safety conditions are specified in terms of forward invariance of a set, input-to-state safety (ISSf) conditions are specified in terms of forward invariance of a slightly larger set. In this context, invariance of the larger set implies that the states stay either inside or very close to the smaller safe set; and this closeness is bounded by the magnitude of the disturbances. The main contribution of the letter is the methodology used for obtaining a valid ISSf-CBF, given a control barrier function (CBF). The associated universal control law will also be provided. Towards the end, we will study unified quadratic programs (QPs) that combine control Lyapunov functions (CLFs) and ISSf-CBFs in order to obtain a single control law that ensures both safety and stability in systems with input disturbances.Index Terms-Safety critical control, barrier functions, input-to-state safety, autonomous systems. arXiv:1803.03035v3 [math.OC]

This paper presents a methodology for achieving efficient multi-domain underactuated bipedal walking on compliant robots by formally emulating gaits produced by the Spring Loaded Inverted Pendulum (SLIP). With the goal of achieving locomotion that displays phases of double and single support, a hybrid system model is formulated that faithfully represents the full-order dynamics of a compliant walking robot. The SLIP model is used as a basis for constructing human-inspired controllers that yield a dimension reduction through the use of hybrid zero dynamics. This allows for the formulation of an optimization problem that produces hybrid zero dynamics that best represents a SLIP model walking gait, while simultaneously ensuring the proper reduction in dimensionality that can be utilized to produce stable periodic orbits, i.e., walking gaits. The end result is stable robotic walking in simulation and, when implemented on the compliant robot ATRIAS, experimentally realized dynamic multi-domain locomotion.

This paper presents the meta-algorithmic approach used to realize multi-contact walking on the humanoid robot, DURUS. This systematic methodology begins by decomposing human walking into a sequence of distinct events (e.g. heel-strike, toe-strike, and toe push-off). These events are converted into an alternating sequence of domains and guards, resulting in a hybrid system model of the locomotion. Through the use of a direct collocation based optimization framework, a walking gait is generated for the hybrid system model emulating human-like multi-contact walking behaviors -additional constraints are iteratively added and shaped from experimental evaluation to reflect the machine's practical limitations. The synthesized gait is analyzed directly on hardware wherein feedback regulators are introduced which stabilize the walking gait, e.g., modulating foot placement. The end result is an energyoptimized walking gait that is physically implementable on hardware. The novelty of this work lies in the creation of a systematic approach for developing dynamic walking gaits on 3D humanoid robots: from formulating the hybrid system model to gait optimization to experimental validation refined to produce multi-contact 3D walking in experiment.

This paper describes a torque control scheme unifying feedback PD control and feed-forward impedance control to realize human-inspired walking on a novel planar footed bipedal robot: AMBER2. It starts with high fidelity modeling of the robot including nonlinear dynamics, motor model, and impact dynamics. Human data is then used by an optimization algorithm to produce a human-like gait that can be implemented on the robot. To realize the bipedal walking, first a PD controller is utilized to track the optimized trajectory. Next, impedance control parameters are estimated from the experimental data. Finally, the unified PD, impedance torque control law is experimentally realized on the bipedal robot AMBER2. Through the evidence of sustainable and unsupported walking on AMBER2 showing high consistency with the simulated gait, the feasibility of AMBER2 walking scheme will be verified.

Over the decades, kinematic controllers have proven to be practically useful for applications like set-point and trajectory tracking in robotic systems. To this end, we formulate a novel safety-critical paradigm for kinematic control in this paper. In particular, we extend the methodology of control barrier functions (CBFs) to kinematic equations governing robotic systems. We demonstrate a purely kinematic implementation of a velocity-based CBF, and subsequently introduce a formulation that guarantees safety at the level of dynamics. This is achieved through a new form CBFs that incorporate kinetic energy with the classical forms, thereby minimizing model dependence and conservativeness. The approach is then extended to underactuated systems. This method and the purely kinematic implementation are demonstrated in simulation on two robotic platforms: a 6-DOF robotic manipulator, and a cart-pole system.

Model based controllers, by virtue of their dependence on a specific model, are highly sensitive to imperfections in model parameter estimation leading to undesirable behaviors, especially in robots that undergo impacts. With the goal of quantifying the effect of model imperfection on the resulting output behavior from a control Lyapunov function (CLF) based controller, we formally derive a measure for model parameter mismatch and show that a bounded measure leads to an ultimate bound on the CLF. This is also extended to the discrete map by introducing an impact measure. The measure is controller and path dependent, and not just parameter dependent, thereby differentiating it from existing methods. More specifically, if traditional methods yield ultimate boundedness for a bounded parameter uncertainty, the proposed "measure" uses the notion of Input to State Stability (ISS) criterion to establish stability of model based controllers. The main result of this paper establishes that the proposed CLF based controller is parameter to state stable (PSS) for a class of robotic hybrid systems-systems with impulsive effects. These formal results motivate the construction of a robust controller-combining a computed torque term with a traditional PD term-that yields stricter convergence rates and bounds on the errors. This is demonstrated on the bipedal robot AMBER with a modeling error 30%, wherein the stability of the proposed controller is verified in simulation.

Implementing state-based parameterized periodic trajectories on complex robotic systems, e.g., humanoid robots, can lead to instability due to sensor noise exacerbated by dynamic movements. As a means of understanding this phenomenon, and motivated by field testing on the humanoid robot DURUS, this paper presents sufficient conditions for the boundedness of hybrid periodic orbits (i.e., boundedness of walking gaits) for time dependent control Lyapunov functions. In particular, this paper considers virtual constraints that yield hybrid zero dynamics with desired outputs that are a function of time or a state-based phase variable. If the difference between the phase variable and time is bounded, we establish exponential boundedness to the zero dynamics surface. These results are extended to hybrid dynamical systems, establishing exponential boundedness of hybrid periodic orbits, i.e., we show that stable walking can be achieved through time-based implementations of state-based virtual constraints. These results are verified on the bipedal humanoid robot DURUS both in simulation and experimentally; it is demonstrated that a close match between time based tracking and state based tracking can be achieved as long as there is a close match between the time and phase based desired output trajectories.

This paper presents a novel optimal control strategy combining control Lyapunov function (CLF) based quadratic programs with impedance control, with the goal of improving both tracking performance and the stability of controllers implemented on transfemoral prosthesis. CLF based quadratic programs have the inherent capacity to optimally track a desired trajectory. This property is used in congruence with impedance control-implemented as a feedforward term-to realize significantly small tracking errors, while simultaneously yielding bipedal walking that is both stable and robust to disturbances. Moreover, instead of experimentally validating this on human subjects, a virtual prosthesis is attached to a robotic testbed, AMBER. The authors claim that the walking of AMBER is human like and therefore form a suitable substitute to human subjects on which a prosthetic control can be tested. Based on this idea, the proposed controller was first verified in simulation, then tested on the physical robot AMBER. The results indicate improved tracking performance, stability, and robustness to unknown disturbances.