This paper presents \ul{CogRun}, a framework that enables safety-critical ground robots to perform cognitively-grounded runtime learning entirely on edge-AI devices in unknown physical environments, without prior maps or perceptual knowledge. CogRun consists of three components: a Learning-Agent, a Rational-Agent, and a Coordinator. The Learning-Agent is novel in cognitive-neural learning architecture, which featurs dedicated replay buffers, cognition-driven experience sampling, and a safety-aware action blending of actor-critic reinforcement learning (RL) with instance-based learning (IBL). The Rational-Agent is a non-learning module that complements the Learning-Agent by exclusively handling safety-critical functions, while the Coordinator manages interactions between the two agents to promote safe and efficient runtime learning. CogRun's full autonomy stack (i.e., perception, learning, and control) on edge-AI devices eliminates dependence on wireless communications, enabling broader applications in challenging environments with limited or no connectivity. Experiments on a quadruped robot in real-world wild forests and on an off-road autonomous vehicle in a simulated wild forest demonstrate that CogRun enables safe and efficient runtime learning, allowing robots to safely and continuously interact with the physical world for enhancing task performance in complex, unknown environments.

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