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Safe and efficient collaboration among multiple robots in unstructured environments is increasingly critical in the era of Industry 4.0. However, achieving robust and autonomous collaboration among humans and other robots requires modern robotic systems to have effective proximity perception and reactive obstacle avoidance. In this paper, we propose a novel methodology for reactive whole-body obstacle avoidance that ensures conflict-free robot-robot interactions even in dynamic environment. Unlike existing approaches based on Jacobian-type, sampling based or geometric techniques, our methodology leverages the latest deep learning advances and topological manifold learning, enabling it to be readily generalized to other problem settings with high computing efficiency and fast graph traversal techniques. Our approach allows a robotic arm to proactively avoid obstacles of arbitrary 3D shapes without direct contact, a significant improvement over traditional industrial cobot settings. To validate our approach, we implement it on a robotic platform consisting of dual 6-DoF robotic arms with optimized proximity sensor placement, capable of working collaboratively with varying levels of interference. Specifically, one arm performs reactive whole-body obstacle avoidance while achieving its pre-determined objective, while the other arm emulates the presence of a human collaborator with independent and potentially adversarial movements. Our methodology provides a robust and effective solution for safe human-robot collaboration in non-stationary environments.