Control tasks with safety requirements under high levels of model uncertainty are increasingly common. Machine learning techniques are frequently used to address such tasks, typically by leveraging model error bounds to specify robust constraint-based safety filters. However, if the learned model uncertainty is very high, the corresponding filters are potentially invalid, meaning no control input satisfies the constraints imposed by the safety filter. While most works address this issue by assuming some form of safe backup controller, ours tackles it by collecting additional data on the fly using a Gaussian process bandit-type algorithm. We combine a control barrier function with a learned model to specify a robust certificate that ensures safety if feasible. Whenever infeasibility occurs, we leverage the control barrier function to guide exploration, ensuring the collected data contributes toward the closed-loop system safety. By combining a safety filter with exploration in this manner, our method provably achieves safety in a general setting that does not require any prior model or backup controller, provided that the true system lies in a reproducing kernel Hilbert space. To the best of our knowledge, it is the first safe learning-based control method that achieves this.

Safe navigation of cluttered environments is a critical challenge in robotics. It is typically approached by separating the planning and tracking problems, with planning executed on a reduced order model to generate reference trajectories, and control techniques used to track these trajectories on the full order dynamics. Inevitable tracking error necessitates robustification of the nominal plan to ensure safety; in many cases, this is accomplished via worst-case bounding, which ignores the fact that some trajectories of the planning model may be easier to track than others. In this work, we present a novel method leveraging massively parallel simulation to learn a dynamic tube representation, which characterizes tracking performance as a function of actions taken by the planning model. Planning model trajectories are then optimized such that the dynamic tube lies in the free space, allowing a balance between performance and safety to be traded off in real time. The resulting Dynamic Tube MPC is applied to the 3D hopping robot ARCHER, enabling agile and performant navigation of cluttered environments, and safe collision-free traversal of narrow corridors.

The deployment of robotic systems in real world environments requires the ability to quickly produce paths through cluttered, non-convex spaces. These planned trajectories must be both kinematically feasible (i.e., collision free) and dynamically feasible (i.e., satisfy the underlying system dynamics), necessitating a consideration of both the free space and the dynamics of the robot in the path planning phase. In this work, we explore the application of reachable Bézier polytopes as an efficient tool for generating trajectories satisfying both kinematic and dynamic requirements. Furthermore, we demonstrate that by offloading specific computation tasks to the GPU, such an algorithm can meet tight real time requirements. We propose a layered control architecture that efficiently produces collision free and dynamically feasible paths for nonlinear control systems, and demonstrate the framework on the tasks of 3D hopping in a cluttered environment.

Humanoid robots have great potential for real-world applications due to their ability to operate in environments built for humans, but their deployment is hindered by the challenge of controlling their underlying high-dimensional nonlinear hybrid dynamics. While reduced-order models like the Hybrid Linear Inverted Pendulum (HLIP) are simple and computationally efficient, they lose whole-body expressiveness. Meanwhile, recent advances in Contact-Implicit Model Predictive Control (CI-MPC) enable robots to plan through multiple hybrid contact modes, but remain vulnerable to local minima and require significant tuning. We propose a control framework that combines the strengths of HLIP and CI-MPC. The reduced-order model generates a nominal gait, while CI-MPC manages the whole-body dynamics and modifies the contact schedule as needed. We demonstrate the effectiveness of this approach in simulation with a novel 24 degree-of-freedom humanoid robot: Achilles. Our proposed framework achieves rough terrain walking, disturbance recovery, robustness under model and state uncertainty, and allows the robot to interact with obstacles in the environment, all while running online in real-time at 50 Hz.

This paper proposes a safety-critical locomotion control framework employed for legged robots exploring through infeasible path in obstacle-rich environments. Our research focus is on achieving safe and robust locomotion where robots confront unavoidable obstacles en route to their designated destination. Through the utilization of outcomes from physical interactions with unknown objects, we establish a hierarchy among the safety-critical conditions avoiding the obstacles. This hierarchy enables the generation of a safe reference trajectory that adeptly mitigates conflicts among safety conditions and reduce the risk while controlling the robot toward its destination without additional motion planning methods. In addition, robust bipedal locomotion is achieved by utilizing the Hybrid Linear Inverted Pendulum model, coupled with a disturbance observer addressing a disturbance from the physical interaction.

Achieving human-like dexterity is a longstanding challenge in robotics, in part due to the complexity of planning and control for contact-rich systems. In reinforcement learning (RL), one popular approach has been to use massively-parallelized, domain-randomized simulations to learn a policy offline over a vast array of contact conditions, allowing robust sim-to-real transfer. Inspired by recent advances in real-time parallel simulation, this work considers instead the viability of online planning methods for contact-rich manipulation by studying the well-known in-hand cube reorientation task. We propose a simple architecture that employs a sampling-based predictive controller and vision-based pose estimator to search for contact-rich control actions online. We conduct thorough experiments to assess the real-world performance of our method, architectural design choices, and key factors for robustness, demonstrating that our simple sampling-based approach achieves performance comparable to prior RL-based works. Supplemental material: https://caltech-amber.github.io/drop.

Robust bipedal locomotion in exoskeletons requires the ability to dynamically react to changes in the environment in real time. This paper introduces the hybrid data-driven predictive control (HDDPC) framework, an extension of the data-enabled predictive control, that addresses these challenges by simultaneously planning foot contact schedules and continuous domain trajectories. The proposed framework utilizes a Hankel matrix-based representation to model system dynamics, incorporating step-to-step (S2S) transitions to enhance adaptability in dynamic environments. By integrating contact scheduling with trajectory planning, the framework offers an efficient, unified solution for locomotion motion synthesis that enables robust and reactive walking through online replanning. We validate the approach on the Atalante exoskeleton, demonstrating improved robustness and adaptability.

Modern autopilot systems are prone to sensor attacks that can jeopardize flight safety. To mitigate this risk, we proposed a modular solution: the secure safety filter, which extends the well-established control barrier function (CBF)-based safety filter to account for, and mitigate, sensor attacks. This module consists of a secure state reconstructor (which generates plausible states) and a safety filter (which computes the safe control input that is closest to the nominal one). Differing from existing work focusing on linear, noise-free systems, the proposed secure safety filter handles bounded measurement noise and, by leveraging reduced-order model techniques, is applicable to the nonlinear dynamics of drones. Software-in-the-loop simulations and drone hardware experiments demonstrate the effectiveness of the secure safety filter in rendering the system safe in the presence of sensor attacks.

Robot learning has produced remarkably effective "black-box" controllers for complex tasks such as dynamic locomotion on humanoids. Yet ensuring dynamic safety, i.e., constraint satisfaction, remains challenging for such policies. Reinforcement learning (RL) embeds constraints heuristically through reward engineering, and adding or modifying constraints requires retraining. Model-based approaches, like control barrier functions (CBFs), enable runtime constraint specification with formal guarantees but require accurate dynamics models. This paper presents SHIELD, a layered safety framework that bridges this gap by: (1) training a generative, stochastic dynamics residual model using real-world data from hardware rollouts of the nominal controller, capturing system behavior and uncertainties; and (2) adding a safety layer on top of the nominal (learned locomotion) controller that leverages this model via a stochastic discrete-time CBF formulation enforcing safety constraints in probability. The result is a minimally-invasive safety layer that can be added to the existing autonomy stack to give probabilistic guarantees of safety that balance risk and performance. In hardware experiments on an Unitree G1 humanoid, SHIELD enables safe navigation (obstacle avoidance) through varied indoor and outdoor environments using a nominal (unknown) RL controller and onboard perception.

A key source of brittleness for robotic systems is the presence of model uncertainty and external disturbances. Most existing approaches to robust control either seek to bound the worst-case disturbance (which results in conservative behavior), or to learn a deterministic dynamics model (which is unable to capture uncertain dynamics or disturbances). This work proposes a different approach: training a state-conditioned generative model to represent the distribution of error residuals between the nominal dynamics and the actual system. In particular we introduce the Online Risk-Informed Optimization controller (ORIO), which uses Discrete-Time Control Barrier Functions, combined with a learned, generative disturbance model, to ensure the safety of the system up to some level of risk. We demonstrate our approach in simulations and hardware, and show that our method can learn a disturbance model that is accurate enough to enable risk-sensitive control of a quadrotor flying aggressively with an unmodelled slung load. We use a conditional variational autoencoder (CVAE) to learn a state-conditioned dynamics residual distribution, and find that the resulting controller can run at 100Hz on an embedded computer and exhibits less conservative behavior while retaining theoretical safety properties.

This paper presents a safety-critical approach to the coordination of robots in dynamic environments. To this end, we leverage control barrier functions (CBFs) with the forward reachable set to guarantee the safe coordination of the robots while preserving a desired trajectory via a layered controller. The top-level planner generates a safety-ensured trajectory for each agent, accounting for the dynamic constraints in the environment. This planner leverages high-order CBFs based on the forward reachable set to ensure safety-critical coordination control, i.e., guarantee the safe coordination of the robots during locomotion. The middle-level trajectory planner employs single rigid body (SRB) dynamics to generate optimal ground reaction forces (GRFs) to track the safety-ensured trajectories from the top-level planner. The whole-body motions to adhere to the optimal GRFs while ensuring the friction cone condition at the end of each stance leg are generated from the low-level controller. The effectiveness of the approach is demonstrated through simulation and hardware experiments.

This paper presents a safety-critical control framework tailored for quadruped robots equipped with a roller arm, particularly when performing locomotive tasks such as autonomous robotic inspection in complex, multi-tiered environments. In this study, we consider the problem of operating a quadrupedal robot in distillation columns, locomoting on column trays and transitioning between these trays with a roller arm. To address this problem, our framework encompasses the following key elements: 1) Trajectory generation for seamless transitions between columns, 2) Foothold re-planning in regions deemed unsafe, 3) Safety-critical control incorporating control barrier functions, 4) Gait transitions based on safety levels, and 5) A low-level controller. Our comprehensive framework, comprising these components, enables autonomous and safe locomotion across multiple layers. We incorporate reduced-order and full-body models to ensure safety, integrating safety-critical control and footstep re-planning approaches. We validate the effectiveness of our proposed framework through practical experiments involving a quadruped robot equipped with a roller arm, successfully navigating and transitioning between different levels within the column tray structure.

Successfully achieving bipedal locomotion remains challenging due to real-world factors such as model uncertainty, random disturbances, and imperfect state estimation. In this work, we propose a novel metric for locomotive robustness – the estimated size of the hybrid forward invariant set associated with the step-to-step dynamics. Here, the forward invariant set can be loosely interpreted as the region of attraction for the discrete-time dynamics. We illustrate the use of this metric towards synthesizing nominal walking gaits using a simulation-in-the-loop learning approach. Further, we leverage discrete-time barrier functions and a sampling-based approach to approximate sets that are maximally forward invariant. Lastly, we experimentally demonstrate that this approach results in successful locomotion for both flat-foot walking and multi-contact walking on the Atalante lower-body exoskeleton.

We study the design of robust and agile controllers for hybrid underactuated systems. Our approach breaks down the task of creating a stabilizing controller into: 1) learning a mapping that is invariant under optimal control, and 2) driving the actuated coordinates to the output of that mapping. This approach, termed Zero Dynamics Policies, exploits the structure of underactuation by restricting the inputs of the target mapping to the subset of degrees of freedom that cannot be directly actuated, thereby achieving significant dimension reduction. Furthermore, we retain the stability and constraint satisfaction of optimal control while reducing the online computational overhead. We prove that controllers of this type stabilize hybrid underactuated systems and experimentally validate our approach on the 3D hopping platform, ARCHER. Over the course of 3000 hops the proposed framework demonstrates robust agility, maintaining stable hopping while rejecting disturbances on rough terrain.

Realizing bipedal locomotion on humanoid robots with point feet is especially challenging due to their highly underactuated nature, high degrees of freedom, and hybrid dynamics resulting from impacts. With the goal of addressing this challenging problem, this paper develops a control framework for realizing dynamic locomotion and implements it on a novel point foot humanoid: ADAM. To this end, we close the loop between Hybrid Zero Dynamics (HZD) and Hybrid linear inverted pendulum (HLIP) based step length regulation. To leverage the full-order hybrid dynamics of the robot, walking gaits are first generated offline by utilizing HZD. These trajectories are stabilized online through the use of a HLIP based regulator. Finally, the planned trajectories are mapped into the full-order system using a task space controller incorporating inverse kinematics. The proposed method is verified through numerical simulations and hardware experiments on the humanoid robot ADAM marking the first humanoid point foot walking. Moreover, we experimentally demonstrate the robustness of the realized walking via the ability to track a desired reference speed, robustness to pushes, and locomotion on uneven terrain.

This paper proposes a comprehensive framework designed for the autonomous inspection of complex environments, with a specific focus on multi-tiered settings such as distillation column trays. Leveraging quadruped robots equipped with roller arms, and through the use of onboard perception, we integrate essential motion components including: locomotion, safe and dynamic transitions between trays, and intermediate motions that bridge a variety of motion primitives. Given the slippery and confined nature of column trays, it is critical to ensure safety of the robot during inspection, therefore we employ a safety filter and footstep re-planning based upon control barrier function representations of the environment. Our framework integrates all system components into a state machine encoding the developed safety-critical planning and control elements to guarantee safety-critical autonomy, enabling autonomous and safe navigation and inspection of distillation columns. Experimental validation in an environment, consisting of industrial-grade chemical distillation trays, highlights the effectiveness of our multi-layered architecture.

Conventional approaches to grasp planning re- quire perfect knowledge of an object’s pose and geometry. Uncertainties in these quantities induce uncertainties in the quality of planned grasps, which can lead to failure. Classically, grasp robustness refers to the ability to resist external disturbances after grasping an object. In contrast, this work studies robustness to intrinsic sources of uncertainty like object pose or geometry affecting grasp planning before execution. To do so, we develop a novel analytic theory of grasping that reasons about this intrinsic robustness by characterizing the effect of friction cone uncertainty on a grasp’s force closure status. We apply this result in two ways. First, we analyze the theoretical guarantees on intrinsic robustness of two grasp metrics in the literature, the classical Ferrari-Canny metric and more recent min-weight metric. We validate these results with hardware trials that compare grasps synthesized with and without robustness guarantees, showing a clear improvement in success rates. Second, we use our theory to develop a novel analytic notion of probabilistic force closure, which we show can generate unique, uncertainty-aware grasps in simulation.

Exoskeleton locomotion must be robust while being adaptive to different users with and without payloads. To address these challenges, this work introduces a data-driven predictive control (DDPC) framework to synthesize walking gaits for lower-body exoskeletons, employing Hankel matrices and a state transition matrix for its data-driven model. The proposed approach leverages DDPC through a multi-layer architecture. At the top layer, DDPC serves as a planner employing Hankel matrices and a state transition matrix to generate a data-driven model that can learn and adapt to varying users and payloads. At the lower layer, our method incorporates inverse kinematics and passivity-based control to map the planned trajectory from DDPC into the full-order states of the lower-body exoskeleton. We validate the effectiveness of this approach through numerical simulations and hardware experiments conducted on the Atalante lower-body exoskeleton with different payloads. Moreover, we conducted a comparative analysis against the model predictive control (MPC) framework based on the reduced-order linear inverted pendulum (LIP) model. Through this comparison, the paper demonstrates that DDPC enables robust bipedal walking at various velocities while accounting for model uncertainties and unknown perturbations.

Estimating regions of attraction (ROAs) of dynamical systems is critical for understanding the operational bounds within which a system will converge to a desired state. In this paper, we introduce a neural network-based approach to approximating ROAs that leverages labeled data generated by offline sampling and simulation of initial conditions, with labels determined by flow membership in an "explicit region of attraction." This framework is designed to estimate ROAs with a level of precision suitable for integration into a motion primitive transition framework as conditions to switch between candidate primitive behaviors. To account for gaps between the simulated environment and the real world, online learning is employed; this refines the offline-learned model of the ROA based on observed discrepancies between predicted and actual system behaviors. We validate this methodology on a quadrupedal robot, demonstrating that our ROA estimates can effectively model regions of attraction for a high-dimensional system. We show this for multiple primitive behaviors and in environments different from the training data. The outcomes highlight the usefulness of our method in estimating regions of attraction and informing transition conditions between primitive behaviors.

The well-known quote from George Box states that: “All models are wrong, but some are useful.” To develop more useful models, we quantify the inaccuracy with which a given model represents a system of interest, so that we may leverage this quantity to facilitate controller synthesis and verification. Specifically, we develop a procedure that identifies a sim2real gap that holds with a minimum probability. Augmenting the nominal model with our identified sim2real gap produces an uncertain model which we prove is an accurate representor of system behavior. We leverage this uncertain model to synthesize and verify a controller in simulation using a probabilistic verification approach. This pipeline produces controllers with an arbitrarily high probability of realizing desired safe behavior on system hardware without requiring hardware testing except for those required for sim2real gap identification. We also showcase our procedure working on two hardware platforms - the Robotarium and a quadruped.

We study automated test generation for testing discrete decision-making modules in autonomous systems. Linear temporal logic is used to encode the system specification - requirements of the system under test - and the test specification, which is unknown to the system and describes the desired test behavior. The reactive test synthesis problem is to find constraints on system actions such that in a test execution, both the system and test specifications are satisfied. To do this, we use the specifications and their corresponding Büchi automata to construct the specification product automaton. Then, a virtual product graph representing all possible test executions of the system is constructed from the transition system and the specification product automaton. The main result of this paper is framing the test synthesis problem as a multi-commodity network flow optimization. This optimization is used to derive reactive constraints on system actions, which constitute the test environment. The resulting test environment ensures that the system meets the test specification while also satisfying the system specification. We illustrate this framework in simulation using grid world examples and demonstrate it on hardware with the Unitree A1 quadruped, where we test dynamic locomotion behaviors reactively.

Achieving stable hopping has been a hallmark challenge in the field of dynamic legged locomotion. Controlled hopping is notably difficult due to extended periods of under-actuation combined with very short ground phases wherein ground interactions must be modulated to regulate global state. In this work, we explore the use of hybrid nonlinear model predictive control paired with a low-level feedback controller in a multi-rate hierarchy to achieve dynamically stable motions on a novel 3D hopping robot. In order to demonstrate richer behaviors on the manifold of rotations, both the planning and feedback layers must be designed in a geometrically consistent fashion; therefore, we develop the necessary tools to employ Lie group integrators and appropriate feedback controllers. We experimentally demonstrate stable 3D hopping on a novel robot, as well as trajectory tracking and flipping in simulation.

The aim of this work is to define a planner that enables robust legged locomotion for complex multi-agent systems consisting of several holonomically constrained quadrupeds. To this end, we employ a methodology based on behavioral systems theory to model the sophisticated and high-dimensional structure induced by the holonomic constraints. The resulting model is then used in tandem with distributed control techniques such that the computational burden is shared across agents while the coupling between agents is preserved. Finally, this distributed model is framed in the context of a predictive controller, resulting in a robustly stable method for trajectory planning. This methodology is tested in simulation with up to five agents and is further experimentally validated on three A1 quadrupedal robots subject to various uncertainties, including payloads, rough terrain, and push disturbances.

Active lower-limb prostheses could enable more natural assisted locomotion by contributing net positive work through important gait events, such as ankle push-off. This paper uses multi-contact models of locomotion together with force-based nonlinear optimization-based controllers to achieve human-like kinematic behavior, including ankle push-off, on a powered transfemoral prosthesis. In particular, we leverage model-based control approaches for dynamic bipedal robotic walking to develop a systematic method to realize human-like walking on a powered prosthesis that does not require subject- specific tuning. The proposed controller is implemented on a prosthesis for 2 subjects without tuning between subjects, emulating subject-specific human kinematic trends on the prosthesis joints. These experimental results demonstrate that our force- based nonlinear control approach achieves better tracking of human-like kinematic trajectories, with an average RMSE of 0.0223 during weight-bearing, compared to 2 non-force-sensing methods with an average RMSE of 0.0411 and 0.0430.

This paper considers centralized mission-planning for a heterogeneous multi-agent system with the aim of locating a hidden target. We propose a mixed observable setting, consisting of a fully observable state-space and a partially observable environment, using a hidden Markov model. First, we construct rapidly exploring random trees (RRTs) to introduce the mixed observable RRT for finding plausible mission plans giving way-points for each agent. Leveraging this construction, we present a path-selection strategy based on a dynamic programming approach, which accounts for the uncertainty from partial observations and minimizes the expected cost. Finally, we combine the high-level plan with model predictive control algorithms to evaluate the approach on an experimental setup consisting of a quadruped robot and a drone. It is shown that agents are able to make intelligent decisions to explore the area efficiently and locate the target through collaborative actions.

The ability to generate robust walking gaits on bipedal robots is key to their successful realization on hard-ware. To this end, this work extends the method of Hybrid Zero Dynamics (HZD) – which traditionally only accounts for locomotive stability via periodicity constraints under perfect impact events – through the inclusion of the saltation matrix with a view toward synthesizing robust walking gaits. By jointly minimizing the norm of the extended saltation matrix and the torque of the robot directly in the gait generation process, we demonstrate that the synthesized gaits are more robust than gaits generated with either term alone; these results are shown in simulation and on hardware for the AMBER-3M planar biped and the Atalante lower-body exoskeleton (both with and without a human subject). The end result is experimental validation that combining saltation matrices with HZD methods produces more robust bipedal walking in practice.

The functional demands of robotic systems often require completing various tasks or behaviors under the effect of disturbances or uncertain environments. Of increasing interest is the autonomy for dynamic robots, such as multirotors, motor vehicles, and legged platforms. Here, disturbances and environmental conditions can have significant impact on the successful performance of the individual dynamic behaviors, referred to as “motion primitives”. Despite this, robustness can be achieved by switching to and transitioning through suitable motion primitives. This paper contributes such a method by presenting an abstraction of the motion primitive dynamics and a corresponding”motion primitive transfer function”. From this, a mixed discrete and continuous “motion primitive graph” is constructed, and an algorithm capable of online search of this graph is detailed. The result is a framework capable of realizing holistic robustness on dynamic systems. This is experimentally demonstrated for a set of motion primitives on a quadrupedal robot, subject to various environmental and intentional disturbances.

Leveraging recent developments in black-box risk-aware verification, we provide three algorithms that generate probabilistic guarantees on (1) optimality of solutions, (2) recursive feasibility, and (3) maximum controller runtimes for general nonlinear safety-critical finite-time optimal controllers. These methods forego the usual (perhaps) restrictive assumptions required for typical theoretical guarantees, e.g. terminal set calculation for recursive feasibility in Nonlinear Model Predictive Control, or convexification of optimal controllers to ensure optimality. Furthermore, we show that these methods can directly be applied to hardware systems to generate controller guarantees on their respective systems.

This paper presents an online framework for synthesizing agile locomotion for bipedal robots that adapts to unknown environments, modeling errors, and external disturbances. To this end, we leverage step-to-step (S2S) dynamics which has proven effective in realizing dynamic walking on underactuated robots-assuming known dynamics and environments. This paper considers the case of uncertain models and environments and presents a data-driven representation of the S2S dynamics that can be learned via an adaptive control approach that is both data-efficient and easy to implement. The learned S2S controller generates desired discrete foot placement, which is then realized on the full-order dynamics of the bipedal robot by tracking desired outputs synthesized from the given foot placement. The benefits of the proposed approach are twofold. First, it improves the ability of the robot to walk at a given desired velocity when compared to the non-adaptive baseline controller. Second, the data-driven approach enables stable and agile locomotion under the effect of various unknown disturbances: additional unmodeled payload, large robot model errors, external disturbance forces, biased velocity estimation, and sloped terrains. This is demonstrated through in-depth evaluation with a high-fidelity simulation of the bipedal robot Cassie subject to the aforementioned disturbances [1].

This paper presents a safety-critical approach to the coordinated control of cooperative robots locomoting in the presence of fixed (holonomic) constraints. To this end, we leverage control barrier functions (CBFs) to ensure the safe cooperation of the robots while maintaining a desired formation and avoiding obstacles. The top-level planner generates a set of feasible trajectories, accounting for both kinematic constraints between the robots and physical constraints of the environment. This planner leverages CBFs to ensure safety-critical coordination control, i.e., guarantee safety of the collaborative robots during locomotion. The middle-level trajectory planner incorporates interconnected single rigid body (SRB) dynamics to generate optimal ground reaction forces (GRFs) to track the safety-ensured trajectories from the top-level planner while addressing the interconnection dynamics between agents. Distributed low-level controllers generate whole-body motion to follow the prescribed optimal GRFs while ensuring the friction cone condition at each end of the stance legs. The effectiveness of the approach is demonstrated through numerical simulations and experimentally on a pair of quadrupedal robots.

The safety-critical control of robotic systems often must account for multiple, potentially conflicting, safety constraints. This paper proposes novel relaxation techniques to address safety-critical control problems in the presence of conflicting safety conditions. In particular, Control Barrier Functions (CBFs) provide a means to encode safety as constraints in a Quadratic Program (QP), wherein multiple safety conditions yield multiple constraints. However, the QP problem becomes infeasible when the safety conditions cannot be simultaneously satisfied. To resolve this potential infeasibility, we introduce a hierarchy between the safety conditions and employ an additional variable to relax the less important safety conditions (Relaxed-CBF-QP). We also formulate a cascaded structure to achieve smaller violations of lower-priority safety conditions (Hierarchical-CBF-QP). The proposed approach, therefore, ensures the existence of at least one solution to the QP problem with the CBFs while dynamically balancing enforcement of additional safety constraints. Importantly, this paper evaluates the impact of different weighting factors in the Hierarchical-CBF-QP and, due to the sensitivity of these weightings in the observed behavior, proposes a method to determine the weighting factors via a sampling-based technique. The validity of the proposed approach is demonstrated through simulations and experiments on a quadrupedal robot navigating to a goal through regions with different levels of danger.

Many approaches to grasp synthesis optimize analytic quality metrics that measure grasp robustness based on finger placements and local surface geometry. However, generating feasible dexterous grasps by optimizing these metrics is slow, often taking minutes. To address this issue, this paper presents FRoGGeR: a method that quickly generates robust precision grasps using the min-weight metric, a novel, almost-everywhere differentiable approximation of the classical$\epsilon$grasp metric. The min-weight metric is simple and interpretable, provides a reasonable measure of grasp robustness, and admits numerically efficient gradients for smooth optimization. We leverage these properties to rapidly synthesize collision-free robust grasps-typically in less than a second. FRoGGeR can refine the candidate grasps generated by other methods (heuristic, data-driven, etc.) and is compatible with many object representations (SDFs, meshes, etc.), We study FRoGGeR's performance on over 40 objects drawn from the YCB dataset, outperforming a competitive baseline in computation time, feasibility rate of grasp synthesis, and picking success in simulation. We conclude that FRoGGeR is fast: it has a median synthesis time of 0.834s over hundreds of experiments.

This article presents a layered control approach for real-time trajectory planning and control of robust cooperative locomotion by two holonomically constrained quadrupedal robots. A novel interconnected network of reduced-order models, based on the single rigid body (SRB) dynamics, is developed for trajectory planning purposes. At the higher level of the control architecture, two different model predictive control (MPC) algorithms are proposed to address the optimal control problem of the interconnected SRB dynamics: centralized and distributed MPCs. The distributed MPC assumes two local quadratic programs that share their optimal solutions according to a one-step communication delay and an agreement protocol. At the lower level of the control scheme, distributed nonlinear controllers are developed to impose the full-order dynamics to track the prescribed reduced-order trajectories generated by MPCs. The effectiveness of the control approach is verified with extensive numerical simulations and experiments for the robust and cooperative locomotion of two holonomically constrained A1 robots with different payloads on variable terrains and in the presence of disturbances. It is shown that the distributed MPC has a performance similar to that of the centralized MPC, while the computation time is reduced significantly.

We propose a method for training ordinary differential equations by using a control-theoretic Lyapunov condition for stability. Our approach, called LyaNet, is based on a novel Lyapunov loss formulation that encourages the inference dynamics to converge quickly to the correct prediction. Theoretically, we show that minimizing Lyapunov loss guarantees exponential convergence to the correct solution and enables a novel robustness guarantee. We also provide practical algorithms, including one that avoids the cost of backpropagating through a solver or using the adjoint method. Relative to standard Neural ODE training, we empirically find that LyaNet can offer improved prediction performance, faster convergence of inference dynamics, and improved adversarial robustness. Our code is available at https://github.com/ivandariojr/LyapunovLearning.

With the increasing prevalence of complex vision-based sensing methods for use in obstacle identification and state estimation, characterizing environment-dependent measurement errors has become a difficult and essential part of modern robotics. This paper presents a self-supervised learning approach to safety-critical control. In particular, the uncertainty associated with stereo vision is estimated, and adapted online to new visual environments, wherein this estimate is leveraged in a safety-critical controller in a robust fashion. To this end, we propose an algorithm that exploits the structure of stereo-vision to learn an uncertainty estimate without the need for ground-truth data. We then robustify existing Control Barrier Function-based controllers to provide safety in the presence of this uncertainty estimate. We demonstrate the efficacy of our method on a quadrupedal robot in a variety of environments. When not using our method safety is violated. With offline training alone we observe the robot is safe, but overly-conservative. With our online method the quadruped remains safe and conservatism is reduced.

Experimental demonstration of complex robotic behaviors relies heavily on finding the correct controller gains. This painstaking process is often completed by a domain expert, requiring deep knowledge of the relationship between parameter values and the resulting behavior of the system. Even when such knowledge is possessed, it can take significant effort to navigate the nonintuitive landscape of possible parameter combinations. In this work, we explore the extent to which preference-based learning can be used to optimize controller gains online by repeatedly querying the user for their preferences. This general methodology is applied to two variants of control Lyapunov function based nonlinear controllers framed as quadratic programs, which provide theoretical guarantees but are challenging to realize in practice. These controllers are successfully demonstrated both on the planar underactuated biped, AMBER, and on the 3D underactuated biped, Cassie. We experimentally evaluate the performance of the learned controllers and show that the proposed method is repeatably able to learn gains that yield stable and robust locomotion.

This paper presents an online walking synthesis methodology to enable dynamic and stable walking on constrained footholds for underactuated bipedal robots. Our approach modulates the change of angular momentum about the foot-ground contact pivot at discrete impact using pre-impact vertical center of mass (COM) velocity. To this end, we utilize the underactuated Linear Inverted Pendulum (LIP) model for approximating the underactuated walking dynamics to provide the desired post-impact angular momentum for each step. Desired outputs are constructed via online optimization combined with closed-form polynomials and tracked via a quadratic program (QP) based controller. This method is demonstrated on two robots, AMBER and 3D Cassie, for which stable walking behaviors with constrained footholds are realized on flat ground, stairs, and randomly located stepping stones.

The weight, space, and power limitations of small aerial vehicles often prevent the application of modern control techniques without significant model simplifications. Moreover, high-speed agile behavior, such as that exhibited in drone racing, make these simplified models too unreliable for safety-critical control. In this work, we introduce the concept of time-varying backup controllers (TBCs): user-specified maneuvers combined with backup controllers that generate reference trajectories which guarantee the safety of nonlinear systems. TBCs reduce conservatism when compared to traditional backup controllers and can be directly applied to multi-agent coordination to guarantee safety. Theoretically, we provide conditions under which TBCs strictly reduce conservatism, describe how to switch between several TBC's and show how to embed TBCs in a multi-agent setting. Experimentally, we verify that TBCs safely increase operational freedom when filtering a pilot's actions and demonstrate robustness and computational efficiency when applied to decentralized safety filtering of two quadrotors.

Humans are able to negotiate downstep behaviors-both planned and unplanned-with remarkable agility and ease. The goal of this paper is to systematically study the translation of this human behavior to bipedal walking robots, even if the morphology is inherently different. Concretely, we begin with human data wherein planned and unplanned downsteps are taken. We analyze this data from the perspective of reduced-order modelling of the human, encoding the center of mass (CoM) kinematics and contact forces, which allows for the translation of these behaviors into the corresponding reduced-order model of a bipedal robot. We embed the resulting behaviors into the full-order dynamics of a bipedal robot via nonlinear optimization-based controllers. The end result is the demonstration of planned and unplanned downsteps in simulation on an underactuated walking robot.

In this article, we holistically present a hybrid-linear inverted pendulum (H-LIP) based approach for synthesizing and stabilizing 3-D foot-underactuated bipedal walking, with an emphasis on thorough hardware realization. The H-LIP is proposed to capture the essential components of the underactuated and actuated part of the robotic walking. The robot walking gait is thendirectlysynthesized based on the H-LIP. We comprehensively characterize the periodic orbits of the H-LIP and provably derive the stepping stabilization via its step-to-step (S2S) dynamics, which is then utilized to approximate the S2S dynamics of the horizontal state of the center of mass of the robotic walking. The approximation facilities a H-LIP based stepping controller to provide desired step sizes to stabilize the robotic walking. By realizing the desired step sizes, the robot achieves dynamic and stable walking. The approach is fully evaluated in both simulation and experiment on the 3-D underactuated bipedal robot Cassie, which demonstrates dynamic walking behaviors with both high versatility and robustness.

We consider the problem of designing policies for Markov decision processes (MDPs) with dynamic coherent risk objectives and constraints. We begin by formulating the problem in a Lagrangian framework. Under the assumption that the risk objectives and constraints can be represented by a Markov risk transition mapping, we propose an optimization-based method to synthesize Markovian policies that lower-bound the constrained risk-averse problem. We demonstrate that the formulated optimization problems are in the form of difference convex programs (DCPs) and can be solved by the disciplined convex-concave programming (DCCP) framework. We show that these results generalize linear programs for constrained MDPs with total discounted expected costs and constraints. Finally, we illustrate the effectiveness of the proposed method with numerical experiments on a rover navigation problem involving conditional-value-at-risk (CVaR) and entropic-value-at-risk (EVaR) coherent risk measures.

Robotic hopping requires high performance and precision, due to its extreme interactions with the environment. Designing a system that will perform optimally, or even stably, for this motion primitive is a significant challenge. In previous work, it was shown that designing a robot with two springs (one in series and one in parallel with the actuator) could dramatically improve performance. However, selecting these springs was an intricate process since their dynamics were tightly coupled, and accomplished through trial and error. This work presents a general optimization framework for interconnected systems that designs the time-based hopping motion, while also designing the shape of nonlinear springs on the robot to yield efficient hopping. Utilizing this method, hopping motions and spring designs were generated simultaneously and experimentally verified on a novel hopping robot.

Current lower-limb prosthesis control methods are primarily model-independent — lacking formal guarantees of stability, relying largely on heuristic tuning parameters for good performance, and neglecting use of the natural dynamics of the system. Model-dependence for prosthesis controllers is difficult to achieve due to the unknown human dynamics. We build upon previous work which synthesized provably stable prosthesis walking through the use of rapidly exponentially stabilizing control Lyapunov functions (RES-CLFs). This paper utilizes RES-CLFs together with force estimation to construct model-based optimization-based controllers for the prosthesis. These are experimentally realized on hardware with onboard sensing and computation. This hardware demonstration has formal guarantees of stability, utilizes the natural dynamics of the system, and achieves superior tracking to other prosthesis trajectory tracking control methods.

The problem of dynamic locomotion over rough terrain requires both accurate foot placement together with an emphasis on dynamic stability. Existing approaches to this problem prioritize immediate safe foot placement over longer term dynamic stability considerations, or relegate the coordination of foot placement and dynamic stability to heuristic methods. We propose a multi-layered locomotion framework that unifies Control Barrier Functions (CBFs) with Model Predictive Control (MPC) to simultaneously achieve safe foot placement and dynamic stability. Our approach incorporates CBF based safety constraints both in a low frequency kinodynamic MPC formulation and a high frequency inverse dynamics tracking controller. This ensures that safety-critical execution is considered when optimizing locomotion over a longer horizon. We validate the proposed method in a 3D stepping-stone scenario in simulation and experimentally on the ANYmal quadruped platform.

Characterizing what types of exoskeleton gaits are comfortable for users, and understanding the science of walking more generally, require recovering a user’s utility landscape. Learning these landscapes is challenging, as walking trajectories are defined by numerous gait parameters, data collection from human trials is expensive, and user safety and comfort must be ensured. This work proposes the Region of Interest Active Learning (ROIAL) framework, which actively learns each user’s underlying utility function over a region of interest that ensures safety and comfort. ROIAL learns from ordinal and preference feedback, which are more reliable feedback mechanisms than absolute numerical scores. The algorithm’s performance is evaluated both in simulation and experimentally for three non-disabled subjects walking inside of a lower-body exoskeleton. ROIAL learns Bayesian posteriors that predict each exoskeleton user’s utility landscape across four exoskeleton gait parameters. The algorithm discovers both commonalities and discrepancies across users’ gait preferences and identifies the gait parameters that most influenced user feedback. These results demonstrate the feasibility of recovering gait utility landscapes from limited human trials.

We present a trajectory planning and control architecture for bipedal locomotion at a variety of speeds on a highly underactuated and compliant bipedal robot. A library of compliant walking trajectories are planned offline, and stored as compact arrays of polynomial coefficients for tracking online. The control implementation uses a floating-base inverse dynamics controller which generates dynamically consistent feedforward torques to realize walking using information obtained from the trajectory optimization. The effectiveness of the controller is demonstrated in simulation and on hardware for walking both indoors on flat terrain and over unplanned disturbances outdoors. Additionally, both the controller and optimization source code are made available on GitHub.

This paper presents a framework that leverages both control theory and machine learning to obtain stable and robust bipedal locomotion without the need for manual parameter tuning. Traditionally, gaits are generated through trajectory optimization methods and then realized experimentally — a process that often requires extensive tuning due to differences between the models and hardware. In this work, the process of gait realization via hybrid zero dynamics (HZD) based optimization is formally combined with preference-based learning to systematically realize dynamically stable walking. Importantly, this learning approach does not require a carefully constructed reward function, but instead utilizes human pairwise preferences. The power of the proposed approach is demonstrated through two experiments on a planar biped AMBER-3M: the first with rigid point-feet, and the second with induced model uncertainty through the addition of springs where the added compliance was not accounted for in the gait generation or in the controller. In both experiments, the framework achieves stable, robust, efficient, and natural walking in fewer than 50 iterations with no reliance on a simulation environment. These results demonstrate a promising step in the unification of control theory and learning.

Global position control for underactuated bipedal walking is a challenging problem due to the lack of actuation on the feet of the robots. In this paper, we apply the Hybrid-Linear Inverted Pendulum (H-LIP) based stepping on 3D underactuated bipedal robots for global position control. The step-to-step (S2S) dynamics of the H-LIP walking approximates the actual S2S dynamics of the walking of the robot, where the step size is considered as the input. Thus the feedback controller based on the H-LIP approximately controls the robot to behave like the H-LIP, the differences between which stay in an error invariant set. Model Predictive Control (MPC) is applied to the H-LIP for global position control in 3D. The H-LIP stepping then generates desired step sizes for the robot to track. Moreover, turning behavior is integrated with the step planning. The proposed framework is verified on the 3D underactuated bipedal robot Cassie in simulation together with a proof-of-concept experiment.

The increasing complexity of modern robotic systems and the environments they operate in necessitates the formal consideration of safety in the presence of imperfect measurements. In this paper we propose a rigorous framework for safety-critical control of systems with erroneous state estimates. We develop this framework by leveraging Control Barrier Functions (CBFs) and unifying the method of Backup Sets for synthesizing control invariant sets with robustness requirements—the end result is the synthesis of Measurement-Robust Control Barrier Functions (MR-CBFs). This provides theoretical guarantees on safe behavior in the presence of imperfect measurements and improved robustness over standard CBF approaches. We demonstrate the efficacy of this framework both in simulation and experimentally on a Segway platform using an onboard stereo-vision camera for state estimation.

We present a straightforward and efficient way to control unstable robotic systems using an estimated dynamics model. Specifically, we show how to exploit the differentiability of Gaussian Processes to create a state-dependent linearized approximation of the true continuous dynamics that can be integrated with model predictive control. Our approach is compatible with most Gaussian process approaches for system identification, and can learn an accurate model using modest amounts of training data. We validate our approach by learning the dynamics of an unstable system such as a segway with a 7-D state space and 2-D input space (using only one minute of data), and we show that the resulting controller is robust to unmodelled dynamics and disturbances, while state-of-the-art control methods based on nominal models can fail under small perturbations. Code is open sourced at https://github.com/learning-and-control/core.

Artificial potential fields (APFs) and their variants have been a staple for collision avoidance of mobile robots and manipulators for almost 40 years. Its model-independent nature, ease of implementation, and real-time performance have played a large role in its continued success over the years. Control barrier functions (CBFs), on the other hand, are a more recent development, commonly used to guarantee safety for nonlinear systems in real-time in the form of a filter on a nominal controller. In this paper, we address the connections between APFs and CBFs. At a theoretic level, we show that given a broad class of APFs, one can construct a CBF that guarantees safety. Additionally, we prove that CBFs obtained from these APFs have additional beneficial properties and can be applied to nonlinear systems. Practically, we compare the performance of APFs and CBFs in the context of obstacle avoidance on simple illustrative examples and for a quadrotor with unknown dynamics, both in simulation and on hardware using onboard sensing.

Functional autonomous systems often realize complex tasks by utilizing state machines comprised of discrete primitive behaviors and transitions between these behaviors. This architecture has been widely studied in the context of quasi-static and dynamics-independent systems. However, applications of this concept to dynamical systems are relatively sparse, despite extensive research on individual dynamic primitive behaviors, which we refer to as "motion primitives." This paper formalizes a process to determine dynamic-state aware conditions for transitions between motion primitives in the context of safety. The result is framed as a "motion primitive graph" that can be traversed by standard graph search and planning algorithms to realize functional autonomy. To demonstrate this framework, dynamic motion primitives— including standing up, walking, and jumping—and the transitions between these behaviors are experimentally realized on a quadrupedal robot.

This paper presents and experimentally implements three different adaptive and robust adaptive controllers as the first steps toward using model-based controllers for transfemoral prostheses. The goal of this paper is to translate these control methods to the robotic domain, from bipedal robotic walking to prosthesis walking, including a rigorous stability analysis. The human/prosthesis system is first modeled as a two-domain hybrid asymmetric system. An optimization problem is formulated to obtain a stable human-like gait. The proposed controllers are then developed for the combined human/prosthesis model and the optimized reference gait. The stability of all three controllers is proven using the Lyapunov stability theorem, ensuring convergence to the desired gait. The proposed controllers are first verified on a bipedal walking robot as a hybrid human/prosthesis model in simulation. They are then experimentally tested on a treadmill with an able-bodied subject using third iteration of AMBER Prosthetic (AMPRO3), a custom self-contained powered transfemoral prosthesis. Finally, outdoor tests are carried out using AMPRO3 with three test subjects walking on level ground, uphill slopes, and downhill slopes at slope angles of 3° and 8°, to demonstrate walking in different real-world environments.

Robotic Hopping is challenging from the perspective of both modeling the dynamics as well as the mechanical design due to the short period of ground contact in which to actuate on the world. Previous work has demonstrated stable hopping on a moving-mass robot, wherein a single spring was utilized below the body of the robot. This paper finds that the addition of a spring in parallel to the actuator greatly improves the performance of moving mass hopping robots. This is demonstrated through the design of a novel one-dimensional hopping robot. For this robot, a rigorous trajectory optimization method is developed using hybrid systems models with experimentally tuned parameters. Simulation results are used to study the effects of a parallel spring on energetic efficiency, stability, and hopping effort. We find that the double-spring model had 2.5x better energy efficiency than the single-spring model, and was able to hop using 40% less peak force from the actuator. Furthermore, the double-spring model produces stable hopping without the need for stabilizing controllers. These concepts are demonstrated experimentally on a novel hopping robot, wherein hop heights up to 40cm were achieved with comparable efficiency and stability.

Lower leg prostheses could improve the life quality of amputees by increasing comfort and reducing energy to locomote, but currently control methods are limited in modulating behaviors based upon the human's experience. This paper describes the first steps toward learning complex controllers for dynamical robotic assistive devices. We provide the first example of behavioral cloning to control a powered transfemoral prostheses using a Gated Recurrent Unit (GRU) based recurrent neural network (RNN) running on a custom hardware accelerator that exploits temporal sparsity. The RNN is trained on data collected from the original prosthesis controller. The RNN inference is realized by a novel EdgeDRNN accelerator in real-time. Experimental results show that the RNN can replace the nominal PD controller to realize end-to-end control of the AMPRO3 prosthetic leg walking on flat ground and unforeseen slopes with comparable tracking accuracy. EdgeDRNN computes the RNN about 240 times faster than real time, opening the possibility of running larger networks for more complex tasks in the future. Implementing an RNN on this real-time dynamical system with impacts sets the ground work to incorporate other learned elements of the human-prosthesis system into prosthesis control.

This paper systematically decomposes quadrupeds into bipeds to rapidly generate walking gaits, and then recomposes these gaits to obtain quadrupedal locomotion.We begin by decomposing the full-order, nonlinear and hybrid dynamics of a three-dimensional quadrupedal robot, including its continuous and discrete dynamics, into two bipedal systems that are subject to external forces.Using the hybrid zero dynamics (HZD) framework, gaits for these bipedal robots can be rapidly generated (on the order of seconds) along with corresponding controllers.The decomposition is performed in such a way that the bipedal walking gaits and controllers can be composed to yield dynamic walking gaits for the original quadrupedal robot -the result, therefore, is the rapid generation of dynamic quadruped gaits utilizing the full-order dynamics.This methodology is demonstrated through the rapid generation (3.96 seconds on average) of four stepping-in-place gaits and one ambling gait at 0.35 m/s on a quadrupedal robot -the Vision 60, with 36 state variables and 12 control inputs -both in simulation and through outdoor experiments.This suggested a new approach for fast quadrupedal trajectory planning using full-body dynamics, without the need for empirical model simplification, wherein methods from dynamic bipedal walking can be directly applied to quadrupeds.

The control of passive dynamic systems remains a challenging problem in the field of robotics, and insights from their study can inform everything from dynamic behaviors on actuated robots to robotic assistive devices. In this work, we explore the use of flat actuated environments for realizing passive dynamic balancing and locomotion. Specifically, we utilize a novel omnidirectional actuated floor to dynamically stabilize two robotic systems. We begin with an inverted pendulum to demonstrate the ability to control a passive system through an active environment. We then consider a passive bipedal robot wherein dynamically stable periodic walking gaits are generated through an optimization that leverages the actuated floor. The end result is the ability to demonstrate passive dynamic walking experimentally through the use of actuated environments.

This paper details a novel approach to collision avoidance for aerial vehicles that enables high-speed flight in uncertain environments. This framework is applied at the controller level and provides safety regardless of the planner that is used. The method is shown to be robust to state uncertainty and disturbances, and is computed entirely online utilizing the full nonlinear system dynamics. The effectiveness of this method is shown in a high-fidelity simulation of a quadrotor with onboard sensors rapidly and safely exploring a cave environment utilizing a simple planner.

This paper presents a personalized gait optimization framework for lower-body exoskeletons. Rather than optimizing numerical objectives such as the mechanical cost of transport, our approach directly learns from user prefer-ences, e.g., for comfort. Building upon work in preference-based interactive learning, we present the CoSpar algorithm. CoSpar prompts the user to give pairwise preferences between trials and suggest improvements; as exoskeleton walking is a non-intuitive behavior, users can provide preferences more easily and reliably than numerical feedback. We show that CoSpar performs competitively in simulation and demonstrate a prototype implementation of CoSpar on a lower-body exoskeleton to optimize human walking trajectory features. In the experiments, CoSpar consistently found user-preferred parameters of the exoskeleton's walking gait, which suggests that it is a promising starting point for adapting and personalizing exoskeletons (or other assistive devices) to individual users.

This paper proposes a data-driven method for powered prosthesis control that achieves stable walking without the need for additional sensors on the human. The key idea is to extract the nominal gait and the human interaction information from motion capture data, and reconstruct the walking behavior with a dynamic model of the human-prosthesis system. The walking behavior of a human wearing a powered prosthesis is obtained through motion capture, which yields the limb and joint trajectories. Then a nominal trajectory is obtained by solving a gait optimization problem designed to reconstruct the walking behavior observed by motion capture. Moreover, the interaction force profiles between the human and the prosthesis are recovered by simulating the model following the recorded gaits, which are then used to construct a force tube that covers all the interaction force profiles. Finally, a robust Control Lyapunov Function (CLF) Quadratic Programming (QP) controller is designed to guarantee the convergence to the nominal trajectory under all possible interaction forces within the tube. Simulation results show this controller's improved tracking performance with a perturbed force profile compared to other control methods with less model information.

Can we design motion primitives for complex legged systems uniformly for different terrain types without neglecting modeling details? This paper presents a method for rapidly generating quadrupedal locomotion on sloped terrains-from modeling to gait generation, to hardware demonstration. At the core of this approach is the observation that a quadrupedal robot can be exactly decomposed into coupled bipedal robots. Formally, this is represented through the framework of coupled control systems, wherein isolated subsystems interact through coupling constraints. We demonstrate this concept in the context of quadrupeds and use it to reduce the gait planning problem for uneven terrains to bipedal walking generation via hybrid zero dynamics. This reduction method allows for the formulation of a nonlinear optimization problem that leverages low-dimensional bipedal representations to generate dynamic walking gaits on slopes for the full-order quadrupedal robot dynamics. The result is the ability to rapidly generate quadrupedal walking gaits on a variety of slopes. We demonstrate these walking behaviors on the Vision 60 quadrupedal robot; in simulation, via walking on a range of sloped terrains of 13°, 15°, 20°, 25°, and, experimentally, through the successful locomotion of 13° and 20° ~ 25° sloped outdoor grasslands.

Hybrid locomotion, which combines multiple modalities of locomotion within a single robot, enables robots to carry out complex tasks in diverse environments. This paper presents a novel method for planning multi-modal locomotion trajectories using approximate dynamic programming. We formulate this problem as a shortest-path search through a state-space graph, where the edge cost is assigned as optimal transport cost along each segment. This cost is approximated from batches of offline trajectory optimizations, which allows the complex effects of vehicle under-actuation and dynamic constraints to be approximately captured in a tractable way. Our method is illustrated on a hybrid double-integrator, an amphibious robot, and a flying-driving drone, showing the practicality of the approach.

Optimizing lower-body exoskeleton walking gaits for user comfort requires understanding users' preferences over a high-dimensional gait parameter space. However, existing preference-based learning methods have only explored low-dimensional domains due to computational limitations. To learn user preferences in high dimensions, this work presents LINECOSPAR, a human-in-the-loop preference-based framework that enables optimization over many parameters by iteratively exploring one-dimensional subspaces. Additionally, this work identifies gait attributes that characterize broader preferences across users. In simulations and human trials, we empirically verify that LINECOSPAR is a sample-efficient approach for high-dimensional preference optimization. Our analysis of the experimental data reveals a correspondence between human preferences and objective measures of dynamicity, while also highlighting differences in the utility functions underlying individual users' gait preferences. This result has implications for exoskeleton gait synthesis, an active field with applications to clinical use and patient rehabilitation.

In this paper, we present a sequential motion planning and control method for generating somersaults on bipedal robots. The somersault (backflip or frontflip) is considered as a coupling between an axile hopping motion and a rotational motion about the center of mass of the robot; these are encoded by a hopping Spring-loaded Inverted Pendulum (SLIP) model and the rotation of a Flywheel, respectively. We thus present the Flywheel SLIP model for generating the desired motion on the ground phase. In the flight phase, we present a momentum transmission method to adjust the orientation of the lower body based on the conservation of the centroidal momentum. The generated motion plans are realized on the full-dimensional robot via momentum-included task space control. Finally, the proposed method is implemented on a modified version of the bipedal robot Cassie in simulation wherein multiple somersault motions are generated.

Practical use of robots in diverse domains requires programming for, or adapting to, each domain and its unique characteristics. Failure to do so compromises the ability of the robot to achieve task-relevant objectives. Here we describe how the learned terrain reaction force profiles of a hopping robot serve the additional objectives of classifying terrain and quickly learning control strategies to accomplish a jumping task on novel terrain. We show that the reaction forces experienced during closed-loop jumping are sufficient to discriminate between three different terrain types (granular, trampoline, and rigid) when using the learned models as discriminators. Building on this, we show that applying the classification to unknown terrain types leads to faster task completion, where the task objective is to meet a specific jump height. The classification experiments, utilizing real-world jumping data, achieve 95% prediction accuracy. The online learning experiments leverage simulation as there is more control over the terrain properties. Terrain-informed learning achieves the target hop heights more than 2x faster than without terrain knowledge when the prediction is correct, and 1.5x faster when the prediction is incorrect. Thus, applying the closest approximately known terrain knowledge facilitates low shot learning when hopping on unknown terrain.

Dynamic bipedal robot locomotion has achieved remarkable success due in part to recent advances in trajectory generation and nonlinear control for stabilization. A key assumption utilized in both theory and experiments is that the robot’s stance foot always makes no-slip contact with the ground, including at impacts. This assumption breaks down on slippery low-friction surfaces, as commonly encountered in outdoor terrains, leading to failure and loss of stability. In this work, we extend the theoretical analysis and trajectory optimization to account for stick-slip transitions at point foot contact using Coulomb’s friction law. Using AMBER-3M planar biped robot as an experimental platform, we demonstrate for the first time a slippery walking gait which can be stabilized successfully both on a lubricated surface and on a rough no-slip surface. We also study the influence of foot slippage on reducing the mechanical cost of transport, and compare energy efficiency in both numerical simulation and experimental measurement.

Hopping is a highly dynamic motion requiring precise input over brief moments of ground contact in order to achieve desired performance. While this problem has been approached from multiple perspectives, this work provides a comparative analysis of two robot models. The first model uses an actuator to store energy in a spring and release it during the ground phase, while the second uses an actuator to move an additional mass vertically to generate force on the spring. In the first model, analytic expressions are used to find the desired controllers, while trajectory optimization is used in the latter. Orbital stability of each model under the conditions of uncertain damping and poor estimation of the hop height is examined. To this end, Poincaré analysis is used to give a metric of stability in the presence of different initial conditions and parameter uncertainty. Simulations show that the first model converges quickly to a point near the desired height determined by the amount of uncertain damping present. The second model is less robust to uncertainty, but is be made to converge to a desired height with the addition of PD control around the optimal trajectory. This robustness is improved with different gains in the controller. In experiments performed on hardware for the second model, stability is observed through convergence to a periodic orbit within several hops.

The hybrid zero dynamics (HZD) approach has become a powerful tool for the gait planning and control of bipedal robots. This paper aims to extend the HZD methods to address walking, ambling and trotting behaviors on a quadrupedal robot. We present a framework that systematically generates a wide range of optimal trajectories and then provably stabilizes them for the full-order, nonlinear and hybrid dynamical models of quadrupedal locomotion. The gait planning is addressed through a scalable nonlinear programming using direct collocation and HZD. The controller synthesis for the exponential stability is then achieved through the Poincaré sections analysis. In particular, we employ an iterative optimization algorithm involving linear and bilinear matrix inequalities (LMIs and BMIs) to design HZD-based controllers that guarantee the exponential stability of the fixed points for the Poincaré return map. The power of the framework is demonstrated through gait generation and HZD-based controller synthesis for an advanced quadruped robot, - Vision 60, with 36 state variables and 12 control inputs. The numerical simulations as well as real world experiments confirm the validity of the proposed framework.

Future manufacturing environments will see an increased need for cooperation between humans and machines. In this paper we propose a method that allows industrial manipulators to safely operate around humans. This approach guarantees that the manipulator will never collide with human operators while performing its normal tasks. This is done in an near-optimal way by considering how forward reachable sets of human operators grow with time, and by continuously updating these reachable sets based on current position estimates of the operators near the robot. An implicit active set invariance filter is then used to constrain the system-in a minimally invasive way-to stay in the complement of that forward reachable set. We demonstrate this approach in simulation on an industrial robotic arm: the ABB IRB 6640.

Many modern nonlinear control methods aim to endow systems with guaranteed properties, such as stability or safety, and have been successfully applied to the domain of robotics. However, model uncertainty remains a persistent challenge, weakening theoretical guarantees and causing implementation failures on physical systems. This paper develops a machine learning framework centered around Control Lyapunov Functions (CLFs) to adapt to parametric uncertainty and unmodeled dynamics in general robotic systems. Our proposed method proceeds by iteratively updating estimates of Lyapunov function derivatives and improving controllers, ultimately yielding a stabilizing quadratic program model-based controller. We validate our approach on a planar Segway simulation, demonstrating substantial performance improvements by iteratively refining on a base model-free controller.

In this paper, 3D humanoid walking is decoupled into periodic and transitional motion, each of which is decoupled into planar walking in the sagittal and lateral plane. Reduced order models (ROMs), i.e. actuated Spring-loaded Inverted Pendulum (aSLIP) models and Hybrid-Linear Inverted Pendulum (H-LIP) models, are utilized for motion generation on the desired center of mass (COM) dynamics for each type of planar motion. The periodic motion is planned via point foot (underactuated) ROMs for dynamic motion with minimum ankle actuation, while the transitional motion is planned via foot-actuated ROMs for fast and smooth transition. Composition of the planar COM dynamics yields the desired COM dynamics in 3D, which is embedded on the humanoid via control Lyapunov function based Quadratic programs (CLF-QPs). Additionally, the ground reaction force profiles of the aSLIP walking are used as desired references for ground contact forces in the CLF-QPs for smooth domain transitions. The proposed framework is realized on a lower-limb exoskeleton in simulation wherein different walking motions are achieved.

A Hybrid passive Linear Inverted Pendulum (H-LIP) model is proposed for characterizing, stabilizing and composing periodic orbits for 3D underactuated bipedal walking. Specifically, Period-l (P1) and Period -2 (P2) orbits are geometrically characterized in the state space of the H-LIP. Stepping controllers are designed for global stabilization of the orbits. Valid ranges of the gains and their optimality are derived. The optimal stepping controller is used to create and stabilize the walking of bipedal robots. An actuated Spring-loaded Inverted Pendulum (aSLIP) model and the underactuated robot Cassie are used for illustration. Both the aSLIP walking with PI or P2 orbits and the Cassie walking with all 3D compositions of the PI and P2 orbits can be smoothly generated and stabilized from a stepping-in-place motion. This approach provides a perspective and a methodology towards continuous gait generation and stabilization for 3D underactuated walking robots.

This paper presents the first experimental results of crutch-less dynamic walking with paraplegics on a lower-body exoskeleton: ATALANTE, designed by the French start-up company Wandercraft. The methodology used to achieve these results is based on the partial hybrid zero dynamics (PHZD) framework for formally generating stable walking gaits. A direct collocation optimization formulation is used to provide fast and efficient generation of gaits tailored to each patient. These gaits are then implemented on the exoskeleton for three paraplegics. The end result is dynamically stable walking in an exoskeleton without the need for crutches. After a short period of tuning by the engineers and practice by the subjects, each subject was able to dynamically walk across a room of about 10 m up to a speed of 0.15 m/s (0.5 km/h) without the need for crutches or any other kind of assistance.

This paper presents the design and validation of controlling hopping on the 3D bipedal robot Cassie. A spring-mass model is identified from the kinematics and compliance of the robot. The spring stiffness and damping are encapsulated by the leg length, thus actuating the leg length can create and control hopping behaviors. Trajectory optimization via direct collocation is performed on the spring-mass model to plan jumping and landing motions. The leg length trajectories are utilized as desired outputs to synthesize a control Lyapunov function based quadratic program (CLF-QP). Centroidal angular momentum, taking as an addition output in the CLF-QP, is also stabilized in the jumping phase to prevent whole body rotation in the underactuated flight phase. The solution to the CLF-QP is a nonlinear feedback control law that achieves dynamic jumping behaviors on bipedal robots with compliance. The framework presented in this paper is verified experimentally on the bipedal robot Cassie.

This paper develops a control approach with correctness guarantees for the simultaneous operation of lane keeping and adaptive cruise control. The safety specifications for these driver assistance modules are expressed in terms of set invariance. Control barrier functions (CBFs) are used to design a family of control solutions that guarantee the forward invariance of a set, which implies satisfaction of the safety specifications. The CBFs are synthesized through a combination of sum-of-squares program and physics-based modeling and optimization. A real-time quadratic program is posed to combine the CBFs with the performance-based controllers, which can be either expressed as control Lyapunov function conditions or as black-box legacy controllers. In both cases, the resulting feedback control guarantees the safety of the composed driver assistance modules in a formally correct manner. Importantly, the quadratic program admits a closed-form solution that can be easily implemented. The effectiveness of the control approach is demonstrated by simulations in the industry-standard vehicle simulator Carsim.

Hybrid zero dynamics (HZD) has emerged as a popular framework for dynamic walking but has significant implementation difficulties when applied to the high degrees of freedom humanoids. The primary impediment is the process of gait design-it is difficult for optimizers to converge on a viable set of virtual constraints defining a gait. This paper presents a methodology that allows for fast and reliable generation of dynamic robotic walking gaits through the HZD framework, even in the presence of underactuation. Specifically, we describe an optimization formulation that builds upon the novel combination of HZD and direct collocation methods. Furthermore, achieving a scalable implementation required developing a defect-variable substitution formulation to simplify expressions, which ultimately allows us to generate compact analytic Jacobians of the constraints. We experimentally validate our methodology on an underactuated humanoid, DURUS, a spring-legged machine designed to facilitate energy-economical walking. We show that the optimization approach, in concert with the HZD framework, yields dynamic and stable walking gaits in hardware with a total electrical cost of transport of 1.33.

Bipedal robotic running remains a challenging benchmark in the field of control and robotics because of its highly dynamic nature and necessarily underactuated hybrid dynamics. Previous results have achieved bipedal running experimentally with a combination of theoretical results and heuristic application thereof. In particular, formal analysis of the hybrid system stability is given based on a theoretical model, but due to the gap between theoretical concepts and experimental reality, extensive tuning is necessary to achieve experimental success. In this paper, we present a formal approach to bridge this gap, starting from theoretical gait generation to a provably stable control implementation, resulting in bipedal robotic running. We first use a large-scale optimization to generate an energy-efficient running gait, subject to hybrid zero dynamics conditions and feasibility constraints which incorporate practical limitations of the robot model based on physical conditions. The stability of the gait is formally guaranteed in the hybrid system model with an input to state stability (ISS) based control law. This implementation improves the stability under practical control limitations of the system. Finally, the methodology is experimentally realized on the planar spring-legged bipedal robot, DURUS-2D, resulting in sustainable running at 1.75m/s. The paper, therefore, presents a formal method that takes the first step toward bridging the gap between theory and experiment.

The varied and complex dynamics of deformable terrain are significant impediments toward real-world viability of locomotive robotics, particularly for legged machines. We explore vertical jumping on granular media (GM) as a model task for legged locomotion on uncharacterized deformable terrain. By integrating (Gaussian process) GP-based regression and evaluation to estimate ground forcing as a function of state, a one-dimensional jumper acquires the ability to learn forcing profiles exerted by its environment in tandem to achieving its control objective. The GP-based dynamical model initially assumes a baseline rigid, non-compliant surface. As part of an iterative procedure, the optimizer employing this model generates an optimal control to achieve a target jump height while respecting known hardware limitations of the robot model. Trajectory and forcing data recovered from evaluation on the true GM surface model simulation is applied to train the GP, and in turn, provide the optimizer a more richly informed dynamical model of the environment. After three iterations, predicted optimal control trajectories coincide with execution results, within 1.2% jumping height error, as the GP-based approximation converges to the true GM model.

Traversing environments with arbitrary obstacles poses significant challenges for bipedal robots. In some cases, whole body motions may be necessary to maneuver around an obstacle, but most existing footstep planners can only select from a discrete set of predetermined footstep actions; they are unable to utilize the continuum of whole body motion that is truly available to the robot platform. Existing motion planners that can utilize whole body motion tend to struggle with the complexity of large-scale problems. We introduce a planning method, called the “Randomized Possibility Graph”, which uses high-level approximations of constraint manifolds to rapidly explore the “possibility” of actions, thereby allowing lower-level motion planners to be utilized more efficiently. We demonstrate simulations of the method working in a variety of semi-unstructured environments. In this context, “semi-unstructured” means the walkable terrain is flat and even, but there are arbitrary 3D obstacles throughout the environment which may need to be stepped over or maneuvered around using whole body motions.

This paper describes the Robotarium -- a remotely accessible, multi-robot research facility. The impetus behind the Robotarium is that multi-robot testbeds constitute an integral and essential part of the multi-robot research cycle, yet they are expensive, complex, and time-consuming to develop, operate, and maintain. These resource constraints, in turn, limit access for large groups of researchers and students, which is what the Robotarium is remedying by providing users with remote access to a state-of-the-art multi-robot test facility. This paper details the design and operation of the Robotarium and discusses the considerations one must take when making complex hardware remotely accessible. In particular, safety must be built into the system already at the design phase without overly constraining what coordinated control programs users can upload and execute, which calls for minimally invasive safety routines with provable performance guarantees.

Safety Barrier Certificates that ensure collision-free maneuvers for teams of differential flatness-based quadrotors are presented in this paper. Synthesized with control barrier functions, the certificates are used to modify the nominal trajectory in a minimally invasive way to avoid collisions. The proposed collision avoidance strategy complements existing flight control and planning algorithms by providing trajectory modifications with provable safety guarantees. The effectiveness of this strategy is supported both by the theoretical results and experimental validation on a team of five quadrotors.

This work presents the preliminary experimental validation of a systematic prosthetic control strategy on a custom compliant transfemoral prosthesis with the end result being energy efficient 3-dimension (3D) multi-contact prosthetic walking. In particular, with the goal of capturing essential components of realistic amputee-prosthesis system, a 3D asymmetric hybrid system model is presented - this forms the foundation for formal gait design and control construction. Based on this model, a two-step direct collocation optimization method is utilized to design an energy efficient multi-contact prosthetic gait in 3D. The designed gaits are also subject to various practical constraints such as human-likeness constraints and comfortability constraints. For experimental validation, a 3D capable powered transfemoral prosthetic device is custom built so as to be amendable to realizing the designed 3D prosthetic gaits. Differentiating this device from existing powered prosthesis, compliant components are added to the three joints (two pitch joints and one roll joint) for the purpose of energy saving and human-like behaviors. Combining the presented control methodology and the novel hardware design, the end result is experimentally realized 3D multi-contact prosthetic walking with improved energy efficiency compared to other devices and control methods.

This paper presents FROST, an open-source MATLAB toolkit for modeling, trajectory optimization and simulation of hybrid dynamical systems with a particular focus in dynamic locomotion. The design objective of FROST is to provide a unified software environment for developing model-based control and motion planning algorithms for robotic systems whose dynamics is hybrid in nature. In particular, FROST uses directed graphs to describe the underlying discrete structure of hybrid system models, which renders it capable of representing a wide variety of robotic systems. Equipped with a custom symbolic math toolbox in MATLAB using Wolfram Mathematica, one can rapidly prototype the mathematical model of robot kinematics and dynamics and generate optimized code of symbolic expressions to boost the speed of optimization and simulation in FROST. In favor of agile and dynamic behaviors, we utilize virtual constraint based motion planning and feedback controllers for robotic systems to exploit the full-order dynamics of the model. Moreover, FROST provides a fast and tractable framework for planning optimal trajectories of hybrid dynamical systems using advanced direct collocation algorithms. FROST has been successfully used to synthesize dynamic walking in multiple bipedal robots. Case studies of such applications are considered in this paper, wherein different types of walking gaits are generated for two specific humanoid robots and validated in simulation.

Achieving stable bipedal robotic walking on deformable terrain is an open and challenging problem at the intersection of robotics and physics. Ground deformation introduces underactuation; uncertainty in terrain dynamics further complicates dynamical modeling and control methods. This work provides a stability criterion for flat-footed bipedal locomotion and allows model-based control methods to function on homogeneous deformable granular terrain (e.g. sand and dirt). By characterizing static reaction forces from granular materials, in conjunction with granular resistive force theory (RFT), we model and validate a static stability region for the center of mass (CoM) projection of a biped on a granular surface. We show that this stability region approximates the admissible Zero Moment Point (ZMP) region for walking, rendering common Linear Inverted Pendulum Model (LIPM) methods valid with our foot placement strategy. By interpreting the stability region as the maximum reaction moment of the terrain, we formulate walking as a hybrid dynamical system and utilize the partial hybrid zero dynamics (PHZD) based methodology to generate walking gaits. Finally, we experimentally validate both the ZMP and PHZD walking gaits on a planar bipedal robot, showing that the stability region criterion permits stable dynamic walking on homogeneous granular terrain.

This paper presents safety barrier certificates that ensure scalable and provably collision-free behaviors in multirobot systems by modifying the nominal controllers to formally satisfy safety constraints. This is achieved by minimizing the difference between the actual and the nominal controllers subject to safety constraints. The resulting computation of the safety controllers is done through a quadratic programming problem that can be solved in real-time and in this paper, we describe a series of problems of increasing complexity. Starting with a centralized formulation, where the safety controller is computed across all agents simultaneously, we show how one can achieve a natural decentralization whereby individual robots only have to remain safe relative to nearby robots. Conservativeness and existence of solutions as well as deadlock-avoidance are then addressed using a mixture of relaxed control barrier functions, hybrid braking controllers, and consistent perturbations. The resulting control strategy is verified experimentally on a collection of wheeled mobile robots whose nominal controllers are explicitly designed to make the robots collide.

Hybrid zero dynamics (HZD) has emerged as a popular framework for dynamic and underactuated bipedal walking, but has significant implementation difficulties when applied to the high degrees of freedom present in humanoid robots. The primary impediment is the process of gait design-it is difficult for optimizers to converge on a viable set of virtual constraints defining a gait. This paper presents a methodology that allows for the fast and reliable generation of efficient multi-contact robotic walking gaits through the framework of HZD, even in the presence of underactuation. To achieve this goal, we unify methods from trajectory optimization with the control framework of multi-domain hybrid zero dynamics. By formulating a novel optimization problem in the context of direct collocation and generating analytic Jacobians for the constraints, solving the resulting nonlinear program becomes tractable for large-scale nonlinear programming solvers, even for systems as high-dimensional as humanoid robots. We experimentally validated our methodology on the spring-legged prototype humanoid, DURUS, showing that the optimization approach yields dynamic and stable 3D walking gaits.

Humanoid robots are designed with dozens of actuated joints to suit a variety of tasks, but walking controllers rarely make the best use of all of this freedom. We present a framework for maximizing the use of the full humanoid body for the purpose of stable dynamic locomotion, which requires no restriction to a planning template (e.g. LIPM). Using a hybrid zero dynamics (HZD) framework, this approach optimizes a set of outputs which provides requirements for the motion for all actuated links, including arms. These output equations are then rapidly solved by a whole-body inverse-kinematic (IK) solver, providing a set of joint trajectories to the robot. We apply this procedure to a simulation of the humanoid robot, DRC-HUBO, which has over 27 actuators. As a consequence, the resulting gaits swing their arms, not by a user defining swinging motions a priori or superimposing them on gaits post hoc, but as an emergent behavior from optimizing the dynamic gait. We also present preliminary dynamic walking experiments with DRC-HUBO in hardware, thereby building a case that hybrid zero dynamics as augmented by inverse kinematics (HZD+IK) is becoming a viable approach for controlling the full complexity of humanoid locomotion.

This paper presents the methodology used to achieve efficient and dynamic walking behaviors on the prototype humanoid robotics platform, DURUS. As a means of providing a hardware platform capable of these behaviors, the design of DURUS combines highly efficient electromechanical components with “control in the loop” design of the leg morphology. Utilizing the final design of DURUS, a formal framework for the generation of dynamic walking gaits which maximizes efficiency by exploiting the full body dynamics of the robot, including the interplay between the passive and active elements, is developed. The gaits generated through this methodology form the basis of the control implementation experimentally realized on DURUS; in particular, the trajectories generated through the formal framework yield a feedforward control input which is modulated by feedback in the form of regulators that compensate for discrepancies between the model and physical system. The end result of the unified approach to control-informed mechanical design, formal gait design and regulator-based feedback control implementation is efficient and dynamic locomotion on the humanoid robot DURUS. In particular, DURUS was able to demonstrate dynamic locomotion at the DRC Finals Endurance Test, walking for just under five hours in a single day, traveling 3.9 km with a mean cost of transport of 1.61-the lowest reported cost of transport achieved on a bipedal humanoid robot.

This paper explores combining task and manipulation planning for humanoid robots. Existing methods tend to either take prohibitively long to compute for humanoids or artificially limit the physical capabilities of the humanoid platform by restricting the robot's actions to predetermined trajectories. We present a hybrid planning system which is able to scale well for complex tasks without relying on predetermined robot actions. Our system utilizes the hybrid backward-forward planning algorithm for high-level task planning combined with humanoid primitives for standing and walking motion planning. These primitives are designed to be efficiently computable during planning, despite the large amount of complexity present in humanoid robots, while still informing the task planner of the geometric constraints present in the problem. Our experiments apply our method to simulated pick-and-place problems with additional gate constraints impacting navigation using the DRC-HUBO1 robot. Our system is able to solve puzzle-like problems on a humanoid within a matter of minutes.

This work demonstrates fast motion planning for robot locomotion that is optimized for terrain with complex dynamics, specifically, rapid penetration of granular media. Gait planning is critical for many legged locomotion control approaches, but they typically assume rigid ground contact. We aim to extend these planning methods to include terrain dynamics we see in the natural world, like sand and dirt, which can both deform and fluidize. Using an added-mass description of collective grain motion, we formulated a model of hydrostatic and hydrodynamic terrain effects that is both principled and representable with closed-form dynamics. As a result, we present a model and fast optimization formulation which solves accurate motion plans on granular media with tractable solving times (6.4-3.8 seconds). For validation, we optimized open-loop motor trajectories for a testbed jumping robot to jump to a target apex height from a bed a loosely packed poppy seeds, a model granular medium. While jumps optimized for rigid ground were anemic on granular media, terrain-aware trajectories hit within 6% of their target. This demonstrates the potential for robot locomotion which meets practical task demands, all while being aware of the terrain beneath it.

Dynamic humanoid locomotion is a challenging control problem, and running is especially difficult to achieve, given the underactuation inherent to aerial domains. Previous work developed a gait-generating optimization framework for dynamic locomotion in the context of hybrid zero dynamics, producing stable 3D walking on the humanoid hardware platform DURUS. Here, we demonstrate that this optimization method also extends to stable 3D running. Gaits generated from the optimization, which utilizes the dynamics of all 23 degrees of freedom to maximize energy economy, results in stable running in a DURUS simulation model. Notably, the presented running is underactuated in all domains, due to DURUS' spring-legged design. Further, we generate 25 different running gaits, over a range of speeds (1.5-3.0 m/s), to demonstrate the reliability of solving the large-scale nonlinear program. We report statistical performance of the optimization in successfully generating stable running (average computation time: 323 seconds) in an effort to establish a benchmark for large-scale gait generation. We inspected this array of gaits across speeds, noting recognizable trends in optimized strategies from prior studies on lower-order models-e.g., both increased step frequency and step length with speed-along with the first reported cost-of-transport curve for a 3D humanoid running model. We consider this result an important step toward humanoid running on the DURUS hardware platform.

This paper presents a novel method of stabilizing hybrid models of torque-constrained, underactuated walking robots - without using nonlinear gait optimization - by leveraging properties of the mechanics of the robot. At its core, the controller stabilizes the transfer of angular momentum from one leg to the next through continuous-time control coupled with hybrid system models that capture impacts that occur at foot strike. In particular, conservation of angular momentum at impact allows for computation of the exact transfer of momentum as a function of the robot's step length and vertical center of mass velocity just prior to foot impact. This motivates the construction of continuous-time reference trajectories for the robot's step length and vertical center of mass with endpoints corresponding to a desired transfer of angular momentum. Stabilization to these trajectories results in stable walking, as indicated by numeric Poincaré analysis. The controller is implemented in simulation of a five-link, underactuated 3D robot via Model Predictive Control which provides a means of achieving walking under non-trivial actuation limits.

Lower-limb prostheses provide a prime example of cyber-physical systems (CPSs) requiring the synergistic development of sensing, algorithms, and controllers. With a view towards better understanding CPSs of this form, this paper presents a systematic methodology using multidomain hybrid system models and optimization-based controllers to achieve human-like multicontact prosthetic walking on a custom-built prosthesis: AMPRO. To achieve this goal, unimpaired human locomotion data is collected and the nominal multicontact human gait is studied. Inspired by previous work which realized multicontact locomotion on the bipedal robot AMBER2, a hybrid system-based optimization problem utilizing the collected reference human gait as reference is utilized to formally design stable multicontact prosthetic gaits that can be implemented on the prosthesis directly. Leveraging control methods that stabilize bipedal walking robots–control Lyapunov function-based quadratic programs coupled with variable impedance control–an online optimization-based controller is formulated to realize the designed gait in both simulation and experimentally on AMPRO. Improved tracking and energy efficiency are seen when this methodology is implemented experimentally. Importantly, the resulting multicontact prosthetic walking captures the essentials of natural human walking both kinematically and kinetically.

Bipedal robots are prime examples of complex cyber-physical systems (CPS). They exhibit many of the features that make the design and verification of CPS so difficult: hybrid dynamics, large continuous dynamics in each mode (e.g., 10 or more state variables), and nontrivial specifications involving nonlinear constraints on the state variables. In this paper, we propose a two-step approach to formally synthesize control software for bipedal robots so as to enforce specifications by design and thereby generate physically realizable stable walking. In the first step, we design outputs and classical controllers driving these outputs to zero. The resulting controlled system evolves on a lower dimensional manifold and is described by the hybrid zero dynamics governing the remaining degrees of freedom. In the second step, we construct an abstraction of the hybrid zero dynamics that is used to synthesize a controller enforcing the desired specifications to be satisfied on the full order model. Our two step approach is a systematic way to mitigate the curse of dimensionality that hampers the applicability of formal synthesis techniques to complex CPS. Our results are illustrated with simulations showing how the synthesized controller enforces all the desired specifications and offers improved performance with respect to a controller that was utilized to obtain walking experimentally on the bipedal robot AMBER 2.

This demonstration presents an unimpaired subject walking with a custom built self-contained powered transfemoral prosthesis: AMPRO, which is controlled by a novel nonlinear real-time optimization based controller. To achieve the behaviors that will be demonstrated, controllers that have been successfully implemented on bipedal walking robots are translated to the prosthesis with the goal of achieving natural human-like walking while minimizing power consumption. To achieve this goal, we begin by collecting reference human locomotion data via Inertial measurement Units (IMUs). This data forms the basis for an optimization problem that generates virtual constraints for the prosthesis that provably yields walking in simulation. Utilizing methods that have proven successful in generating stable robotic locomotion, control Lyapunov function (CLF) based Quadratic Programs (QPs) are utilized to optimally track the resulting desired trajectories. The parameterization of the trajectories is determined through a combination of on-board sensing on the prosthesis together with IMU data, thereby coupling the actions of the user with the controller. Finally, impedance control is integrated into the QP yielding an optimization based control law that displays remarkable tracking and robustness, outperforming traditional PD and impedance control strategies.

This paper presents experimentally realized bipedal robotic walking using ideal torque controllers via a novel approach termed the ideal model resolved motion method (IM-RMM), where a system's ideal closed-loop dynamics are integrated forward from the actual state of the hardware to provide desired positions and velocity commands to a PD controller. By combining this method with gaits generated using the Human-Inspired Control framework, walking was realized experimentally on the DURUS platform, designed and built by SRI, and achieved with minimal system identification. For comparison, two controllers, one using feedback linearization and one using Control Lyapunov Function based Quadratic Programs (CLF-QP), both realized through IM-RMM, are compared with a benchmark procedure, the Hybrid Zero Dynamics reconstruction, that is shown to provide reliable walking in literature. The results of both simulations and experiments are presented, with the CLF-QP implemented via IM-RMM resulting in the lowest experimental specific energetic cost of transport of cet= 0.63 achieved during sustained walking on the 31.5 kg bipedal robot.

Hybrid zero dynamics (HZD) has emerged as a popular framework for the stable control of bipedal robotic gaits, but typically designing a gait's virtual constraints is a slow and undependable optimization process. To expedite and boost the reliability of HZD gait generation, we borrow methods from trajectory optimization to formulate a smoother and more linear optimization problem. We present a multiple-shooting formulation for the optimization of virtual constraints, combining the stability-friendly properties of HZD with an optimization-conducive problem formulation. To showcase the implications of this recipe for improving gait generation, we use the same process to generate periodic planar walking gaits on two different robot models, and in one case, demonstrate stable walking on the hardware prototype, DURUS-R.

This paper addresses the problem of controlling underactuated bipedal walking robots in the presence of actuator torque saturation. The proposed method synthesizes elements of the Human-Inspired Control (HIC) approach for generating provably-stable walking controllers, rapidly exponentially stabilizing control Lyapunov functions (RES-CLFs) and standard model predictive control (MPC). Specifically, the proposed controller uses feedback linearization to construct a linear control system describing the dynamics of the walking outputs. The input to this linear system is designed to be the solution of a MPC-based Quadratic Program which minimizes the sum of the values of a RES-CLF-describing the walking control objectives-over a finite-time horizon. Future values of the torque constraints are mapped into the linear control system using the Hybrid Zero Dynamics property of HIC and subsequently incorporated in the Quadratic Program. The proposed method is implemented in a rigid-body dynamics simulation and initial experiments with the Durus robot.

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 bases 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 a method for achieving planar multi-phase, multi-contact robotic walking using human inspired control and optimization. The walking presented contains phases with differing degrees of actuation including over-actuated double support, fully-actuated single support, and under-actuated single support via heel lift. An optimization methodology for generating walking gaits using partial hybrid zero dynamics will be presented. It will be shown that this method yields periodic, multi-contact locomotion. Simulation results for the three domain walking under standard Input-Output Linearization control will be presented.

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.

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.

This paper briefly presents the process of formally achieving bipedal robotic walking through controller synthesis inspired by human locomotion. Motivated by the hierarchical control present in humans, we begin by viewing the human as a "black box" and describe outputs, or virtual constraints, that appear to characterize human walking. By considering the equivalent outputs for the bipedal robot, a nonlinear controller can be constructed that drives the outputs of the robot to the outputs of the human; moreover, the parameters of this controller can be optimized so that stable robotic walking is provably achieved while simultaneously producing outputs of the robot that are as close as possible to those of a human. Finally, considering a control Lyapunov function based representation of these outputs allows for the class of controllers that provably achieve stable robotic walking can be greatly enlarged. The end result is the generation of bipedal robotic walking that is remarkably human-like and is experimentally realizable, as evidenced by the implementation of the resulting controllers on multiple robotic platforms.

This paper employs the Human-Inspired Control framework in the formal design, optimization and implementation of controllers for 3D bipedal robotic walking. In this framework, controllers drive the robot to a low-dimensional representation, termed the partial hybrid zero dynamics, which is shaped by the parameters of the outputs describing human locomotion data. The main result of this paper is the use of partial hybrid zero dynamics in an optimization problem to compute physical constraints on the robot, without integrating the dynamics of the system, and while simultaneously yielding provably stable walking controllers for a 3D robot model. Controllers corresponding to various walking speeds are obtained through a second speed regulation optimization, and formal methods are presented which provide smooth transitions between walking speeds. These formal results are demonstrated through simulation and utilized to obtain 3D walking experimentally with the NAO robot.

Impedance control is a common framework for control of lower-limb prosthetic devices. This approach requires choosing many impedance controller parameters. In this paper, we show how to learn these parameters for lower-limb prostheses by observation of unimpaired human walkers. We validate our approach in simulation of a transfemoral amputee, and we demonstrate the performance of the learned parameters in a preliminary experiment with a lower-limb prosthetic device.

This paper demonstrates the process of utilizing human locomotion data to formally design controllers that yield provably stable robotic walking and experimentally realizing these formal methods to achieve dynamically stable bipedal robotic walking on the NAO robot. Beginning with walking data, outputs---or functions of the kinematics---are determined that result in a low-dimensional representation of human locomotion. These same outputs can be considered on a robot, and human-inspired control is used to drive the outputs of the robot to the outputs of the human. An optimization problem is presented that determines the parameters of this controller that provide the best fit of the human data while simultaneously ensuring partial hybrid zero dynamics. The main formal result of this paper is a proof that these same parameters result in a stable hybrid periodic orbit with a fixed point that can be computed in closed form. Thus, starting with only human data we obtain a stable walking gait for the bipedal robot model. These formal results are validated through experimentation: implementing the stable walking found in simulation on NAO results in dynamically stable robotic walking that shows excellent agreement with the simulated behavior from which it was derived.

This paper presents the first steps toward going from human data to formal controller design to experimental realization in the context of underactuated bipedal robots. Specifically, by studying experimental human walking data, we find that specific outputs of the human, i.e., functions of the kinematics, appear to be canonical to walking and are all characterized by a single function of time, termed a human walking function. Using the human outputs and walking function, we design a human-inspired controller that drives the output of the robot to the output of the human as represented by the walking function. The main result of the paper is an optimization problem that determines the parameters of this controller so as to guarantee stable underactuated walking that is as “close” as possible to human walking. This result is demonstrated through the simulation of a physical underactuated 2D bipedal robot, AMBER. Experimentally implementing this control on AMBER through “feed-forward” control, i.e., trajectory tracking, repeatedly results in 5-10 steps.

This paper presents an approach to the development of bipedal robotic control techniques for multiple locomotion behaviors. Insight into the fundamental behaviors of human locomotion is obtained through the examination of experimental human data for walking on flat ground, upstairs and downstairs. Specifically, it is shown that certain outputs of the human, independent of locomotion terrain, can be characterized by a single function, termed the extended canonical human function. Optimized functions of this form are tracked via feedback linearization in simulations of a planar robotic biped walking on flat ground, upstairs and downstairs - these three modes of locomotion are termed “motion primitives.” A second optimization is presented, which yields controllers that evolve the robot from one motion primitive to another - these modes of locomotion are termed “motion transitions.” A final simulation is given, which shows the controlled evolution of a robotic biped as it transitions through each mode of locomotion over a pyramidal staircase.

This work presents human-inspired control strategies required for achieving three motion primitives in walking-flat-ground, uneven terrain and up-slope-in an underactuated physical bipedal robot: AMBER. Formal models and controllers which provably guarantee the stability of walking are developed and verified in the simulation. Computationally tractable conditions are given that allow for the experimental implementation of these formal methods through the closed form approximation of constraints that restrict maximum torque, maximum velocity and ensure proper foot clearance. Considering the special property of the motors used in the robot, i.e., low leakage inductance and high angular speed, we approximate the motor model and translate the formal controllers satisfying these constraints into an efficient voltage-based controller that can be directly implemented on AMBER. The end result is robotic walking on AMBER for the three motion primitives that shows good agreement with the formal results from which it was derived.

This paper presents a method for achieving stable “human-like” running in simulation by using human-inspired control. Data from human running experiments are processed, analyzed and split into the two domains: stance phase and flight phase. By examining this data, we present a set of outputs, i.e., functions of the kinematics, which appear to represent human running; moreover, we show that this output data can be described by the time-solution to a linear spring-mass-damper—termed the canonical locomotion function. This observation motivates the construction of a human-inspired optimization that determines the parameters of the canonical locomotion function that provide the best fit of the human data while simultaneously enforcing partial hybrid zero dynamics, i.e., that the human outputs track the canonical locomotion functions even through impacts. The main result is a method for numerically solving this optimization problem that provably results in stable robotic running. Simulation results are presented that demonstrate the “human-like” robotic running obtained through this procedure.

This paper studies the walking behavior of kneed bipedal robots with knee-lock and knee-bounce, formally demonstrating that if knee-locking results in stable bipedal walking, then small amounts of knee-bounce still will result in a walking gait for the robot. To achieve this result, hybrid system models of bipeds are considered wherein knee-bounce corresponds to Zeno behavior. Using results on Zeno stability, we propose a notion of generalized completion that allows solutions to be carried beyond the Zeno point, i.e., carried beyond knee-bounce. We assume that the completed hybrid system has a periodic orbit when the impacts are perfectly plastic - a plastic periodic orbit, or walking gait with knee-lock. The main result of this paper is that when the assumption of perfectly plastic impacts is relaxed, if the plastic periodic orbit is stable and the Zeno point is Zeno stable, then there exists a periodic orbit in the case of non-plastic impacts, i.e., a Zeno periodic orbit corresponding to walking with knee-bounce. This formal result is applied to a specific example of a bipedal robot with knees.

This paper proposes a cost function constructed from human data, the human-based cost, which is used to gauge the "human-like" nature of robotic walking. This cost function is constructed by utilizing motion capture data from a 9 subject straight line walking experiment. Employing a novel technique to process the data, we determine the times when the number of contact points change during the course of a step which automatically determines the ordering of discrete events or the domain breakdown along with the amount of time spent in each domain. The result is a weighted graph or walking cycle, associated with each of the subjects walking gaits. Finding a weighted cycle that minimizes the cut distance between this collection of graphs produces an optimal or universal domain graph for walking together with an optimal walking cycle. In essence, we find a single domain graph and the time spent in each domain that yields the most "natural" and "human-like" bipedal walking. The human-based cost is then defined as the cut distance from this optimal gait. The main findings of this paper are two-fold: (1) when the human-based cost is computed for subjects in the experiment it detects medical conditions that result in aberrations in their walking, and (2) when the human-based cost is computed for existing robotic models the more human-like walking gaits are correctly identified.

A method is proposed which enables testing of prosthetic devices in simulation. A hybrid model is used to represent human walking—the combination of continuous and discrete dynamics motivates the use of hybrid systems. A human walking experiment is analyzed and mathematical functions on the kinematics of the collected data are found which capture some of the fundamental behaviors associated with human walking. One model is considered in which these behaviors are fully tracked using feedback linearization; the intent of this is to simulate healthy human walking. Then, a second model is considered: this model is assumed to be a human with a transfemoral prosthesis; PD control is used on the prosthesis. All models considered demonstrate locally exponentially stable periodic orbits when simulated for four separate test subjects, or, in other words, the models exhibit stable walking even with a prosthetic lower extremity. The methods used in this paper are a stepping stone toward a process capable of rapidly prototyping potential prosthesis designs and controllers.

The equivalence of the stability of periodic orbits with the stability of fixed points of a Poincaré map is a well-known fact for smooth dynamical systems. In particular, the eigenvalues of the linearization of a Poincaré map can be used to determine the stability of periodic orbits. The main objective of this paper is to study the properties of Poincaré maps for hybrid systems as they relate to the stability of hybrid periodic orbits. The main result is that the properties of Poincaré maps for hybrid systems are fundamentally different from those for smooth systems, especially with respect to the linearization of the Poincaré map and its eigenvalues. In particular, the linearization of any Poincaré map for a smooth dynamical system will have one trivial eigenvalue equal to 1 that does not affect the stability of the orbit. For hybrid systems, the trivial eigenvalues are equal to 0 and the number of trivial eigenvalues is bounded above by dimensionality differences between the different discrete domains of the hybrid system and the rank of the reset maps. Specifically, if n is the minimum dimension of the domains of the hybrid system, then the Poincaré map on a domain of dimension m ≥ n results in at least m-n+1 trivial 0 eigenvalues, with the remaining eigenvalues determining the stability of the hybrid periodic orbit. These results will be demonstrated on a nontrivial multi-domain hybrid system: a planar bipedal robot with knees.