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In what follows, we give some details of content omitted in the paper due to space limit. The supplements are organized as follows. We give some proof of Lemma 1, 2, Proposition 1, Lemma 3, 4, and Theorem 2 in Section A.1 -A.6, respectively. We provide some training details in Section A.13 as well as experiment details and results in Section A.14. We compare polynomial CBFs with NCBF in A.15, compare NCBFs with different activation functions in A.16. A.1 Proof of Lemma 1 We prove by induction on L. If L =1, then x 2 X(S) if the pre-activation input to the (1,j) neuron is nonnegative for all j 2 S1 and nonpositive for all j/2 S1. We have that the pre-activation input is equal to WT1jx+r1j, establishing the result for L =1 .

In CBF-based control, the desired safety properties of the system are mapped to nonnegativity of a CBF, and the control input is chosen to ensure that the CBF remains nonnegative for all time.

In what follows, we give some details of content omitted in the paper due to space limit. The proof approach is based on Nagumo's Theorem, which gives necessary and sufficient conditions Definition 2. Let A be a closed set. The following is a fundamental preliminary result for establishing positive invariance. Proposition 2. F or any x 2 @ D, we have T Lemma 2 is a consequence of Proposition 2. For ease of exposition, we first reproduce the lemma First, suppose that condition (i) holds. Next, suppose that condition (ii) holds.

Abstract-- We present a novel modularized end-to-end framework for legged reactive navigation in complex dynamic environments using a single light detection and ranging (LiDAR) sensor . The system comprises four simulation-trained modules: three reinforcement-learning (RL) policies for locomotion, safety shielding, and navigation, and a transformer-based exteroceptive estimator that processes raw point-cloud inputs. This modular decomposition of complex legged motor-control tasks enables lightweight neural networks with simple architectures, trained using standard RL practices with targeted reward shaping and curriculum design, without reliance on heuristics or sophisticated policy-switching mechanisms. We conduct comprehensive ablations to validate our design choices and demonstrate improved robustness compared to existing approaches in challenging navigation tasks. The resulting reactive safe navigation (REASAN) system achieves fully onboard and real-time reactive navigation across both single-and multi-robot settings in complex environments. We release our training and deployment code at https://github.com/ASIG-X/REASAN Legged robots offer distinct advantages given their universal mobility, with expanding application scenarios ranging over search and rescue, logistics, entertainment, industrial inspection, and forestry inventories [1]-[4]. Recent advances in quadrupedal locomotion have demonstrated remarkable performance, particularly, in handling complex static terrains [5]-[7].

The development of athletic humanoid robots has gained significant attention as advances in actuation, sensing, and control enable increasingly dynamic, real-world capabilities. RoboCup, an international competition of fully autonomous humanoid robots, provides a uniquely challenging benchmark for such systems, culminating in the long-term goal of competing against human soccer players by 2050. This paper presents the hardware and software innovations underlying our team's victory in the RoboCup 2024 Adult-Sized Humanoid Soccer Competition. On the hardware side, we introduce an adult-sized humanoid platform built with lightweight structural components, high-torque quasi-direct-drive actuators, and a specialized foot design that enables powerful in-gait kicks while preserving locomotion robustness. On the software side, we develop an integrated perception and localization framework that combines stereo vision, object detection, and landmark-based fusion to provide reliable estimates of the ball, goals, teammates, and opponents. A mid-level navigation stack then generates collision-aware, dynamically feasible trajectories, while a centralized behavior manager coordinates high-level decision making, role selection, and kick execution based on the evolving game state. The seamless integration of these subsystems results in fast, precise, and tactically effective gameplay, enabling robust performance under the dynamic and adversarial conditions of real matches. This paper presents the design principles, system architecture, and experimental results that contributed to ARTEMIS's success as the 2024 Adult-Sized Humanoid Soccer champion.

Ensemble control aims to steer a population of dynamical systems using a shared control input. This paper introduces a constrained ensemble control framework for parameterized, heterogeneous robotic systems operating under state and environmental constraints, such as obstacle avoidance. We develop a moment kernel transform that maps the parameterized ensemble dynamics to the moment system in a kernel space, enabling the characterization of population-level behavior. The state-space constraints, such as polyhedral waypoints to be visited and obstacles to be avoided, are also transformed into the moment space, leading to a unified formulation for safe, large-scale ensemble control. Expressive signal temporal logic specifications are employed to encode complex visit-avoid tasks, which are achieved through a single shared controller synthesized from our constrained ensemble control formulation. Simulation and hardware experiments demonstrate the effectiveness of the proposed approach in safely and efficiently controlling robotic ensembles within constrained environments.

Autonomous ground vehicles (AGVs) must navigate safely in cluttered environments while accounting for complex dynamics and environmental uncertainty. Hamilton-Jacobi Reachability (HJR) offers formal safety guarantees through the computation of forward and backward reachable sets, but its application is hindered by poor scalability in environments with numerous obstacles. In this paper, we present a novel framework called NeuroHJR that leverages Physics-Informed Neural Networks (PINNs) to approximate the HJR solution for real-time obstacle avoidance. By embedding system dynamics and safety constraints directly into the neural network loss function, our method bypasses the need for grid-based discretization and enables efficient estimation of reachable sets in continuous state spaces. We demonstrate the effectiveness of our approach through simulation results in densely cluttered scenarios, showing that it achieves safety performance comparable to that of classical HJR solvers while significantly reducing the computational cost. This work provides a new step toward real-time, scalable deployment of reachability-based obstacle avoidance in robotics.

Autonomous navigation in partially observable environments requires agents to reason beyond immediate sensor input, exploit occlusion, and ensure safety while progressing toward a goal. These challenges arise in many robotics domains, from urban driving and warehouse automation to defense and surveillance. Classical path planning approaches and memoryless reinforcement learning often fail under limited fields of view (FoVs) and occlusions, committing to unsafe or inefficient maneuvers. We propose a hierarchical navigation framework that integrates a Deep Transformer Q-Network (DTQN) as a high-level subgoal selector with a modular low-level controller for waypoint execution. The DTQN consumes short histories of task-aware features, encoding odometry, goal direction, obstacle proximity, and visibility cues, and outputs Q-values to rank candidate subgoals. Visibility-aware candidate generation introduces masking and exposure penalties, rewarding the use of cover and anticipatory safety. A low-level potential field controller then tracks the selected subgoal, ensuring smooth short-horizon obstacle avoidance. We validate our approach in 2D simulation and extend it directly to a 3D Unity-ROS environment by projecting point-cloud perception into the same feature schema, enabling transfer without architectural changes. Results show consistent improvements over classical planners and RL baselines in success rate, safety margins, and time to goal, with ablations confirming the value of temporal memory and visibility-aware candidate design. These findings highlight a generalizable framework for safe navigation under uncertainty, with broad relevance across robotic platforms.