Aaron D. Ames

This paper generalizes the control barrier function framework by replacing scalar-valued functions with matrix-valued ones. Specifically, we develop barrier conditions for safe sets defined by matrix inequalities—both semidefinite and indefinite. Matrix inequalities provide a richer representation of safety constraints, generalizing multiple scalar CBFs and enabling nonsmooth safety constraints, including those arising from spectral conditions. The safety filters constructed from our proposed matrix control barrier functions via semidefinite programming (CBF-SDP) are shown to be continuous. Our matrix formulation naturally provides a continuous safety filter for Boolean-based control barrier functions, notably for disjunctions (OR), without relaxing the safe set. We illustrate the effectiveness of the proposed framework with applications in drone network connectivity maintenance—which relies on non-diagonal matrix-valued constraints—and nonsmooth obstacle avoidance, both in simulations and hardware experiments.

Lyapunov’s theorem provides a fundamental characterization of the stability of dynamical systems. This paper presents a categorical framework for Lyapunov theory, generalizing stability analysis with Lyapunov functions categorically. Core to our approach is the set of axioms underlying a setting for stability , which give the necessary ingredients for “doing Lyapunov theory” in a category of interest. With these minimal assumptions, we define the stability of equilibria, formulate Lyapunov morphisms , and demonstrate that the existence of Lyapunov morphisms is necessary and sufficient for establishing the stability of flows. To illustrate these constructions, we show how classical notions of stability, e.g., for continuous and discrete time dynamical systems, are captured by this categorical framework for Lyapunov theory. Finally, to demonstrate the extensibility of our framework, we illustrate how enriched categories, e.g., Lawvere metric spaces, yield settings for stability enabling one to “do Lyapunov theory” in enriched categories.

Humanoid robots stand at the forefront of robotics, aiming to capture the agility, robustness, and expressivity of human movement in an anthropomorphic form. The locomotion control of humanoids has evolved from classical model–based methods to reinforcement learning powered by large-scale simulation and now to generative models that produce adaptive, whole-body behaviors, propelling humanoids toward operation in real-world environments. This survey positions humanoid control at a turning point, converging toward a unified paradigm of physics-guided generative intelligence that integrates optimization, learning, and predictive reasoning. We identify three core principles linking these paradigms: physics-based modeling, constrained decision-making, and adaptation to uncertainty. Building on these connections, we provide recommendations for researchers and outline open challenges in safety, accessibility, and human-level capability. These directions represent a transformation from engineered stability to intelligent autonomy, laying the groundwork for humanoid generalists capable of operating safely, collaborating naturally, and extending human capability in the open world.

This paper develops certificates that propagate compatibility of multiple control barrier function (CBF) constraints from sampled vertices to their convex hull. Under mild concavity and affinity assumptions, we present three sufficient feasibility conditions under which feasible inputs over the convex hull can be obtained per coordinate, with a common input, or via convex blending. We also describe the associated computational methods, based on interval intersections or an offline linear program (LP). Beyond certifying compatibility, we give conditions under which the quadratic-program (QP) safety filter is affine in the state. This enables explicit implementations via convex combinations of vertex-feasible inputs. Case studies illustrate the results.

Reinforcement learning (RL) has shown promise in generating robust locomotion policies for bipedal robots, but often suffers from tedious reward design and sensitivity to poorly shaped objectives. In this work, we propose a structured reward shaping framework that leverages model-based trajectory generation and control Lyapunov functions (CLFs) to guide policy learning. We explore two model-based planners for generating reference trajectories: a reduced-order linear inverted pendulum (LIP) model for velocity-conditioned motion planning, and a precomputed gait library based on hybrid zero dynamics (HZD) using full-order dynamics. These planners define desired end-effector and joint trajectories, which are used to construct CLF-based rewards that penalize tracking error and encourage rapid convergence. This formulation provides meaningful intermediate rewards, and is straightforward to implement once a reference is available. Both the reference trajectories and CLF shaping are used only during training, resulting in a lightweight policy at deployment. We validate our method both in simulation and through extensive real-world experiments on a Unitree G1 robot. CLF-RL demonstrates significantly improved robustness relative to the baseline RL policy and better performance than a classic tracking reward RL formulation.

Measurements and state estimates are often imperfect in control practice, posing challenges for safety-critical applications, where safety guarantees rely on accurate state information. In the presence of estimation errors, several prior robust control barrier function (R-CBF) formulations have imposed strict conditions on the input. These methods can be overly conservative and can introduce issues such as infeasibility, high control effort, etc. This work proposes a systematic method to improve R-CBFs, and demonstrates its advantages on a tracked vehicle that navigates among multiple obstacles. A primary contribution is a new optimization-based online parameter adaptation scheme that reduces the conservativeness of existing R-CBFs. In order to reduce the complexity of the parameter optimization, we merge several safety constraints into one unified numerical CBF via Poisson’s equation. We further address the dual relative degree issue that typically causes difficulty in vehicle tracking. Experimental trials demonstrate the overall performance improvement of our approach over existing formulations.