Ensuring safety of learning-enabled robotic manipulation across diverse embodiments and tasks still requires significant manual engineering. For collision avoidance, existing approaches typically rely on embodiment-specific learned safety approximations or simplified robot models together with stopping-based fallback controllers that compromise performance. In this paper, we propose an embodiment-agnostic safeguarding approach, X-Safe, which provides formal probabilistic guarantees for collision avoidance without model simplifications or learned safety functions. By operating in the configuration space, our safeguarding method works with different types of robot policies. X-Safe reasons locally around the current configuration to establish safe regions and uses action masking to restrict the policy's support to these safe regions. We demonstrate X-Safe on two embodiments and three policies, both in simulation and on hardware. We observe less than half of the performance degradation compared to state-of-the-art safeguarding, and empirically corroborate our formal collision avoidance guarantees.

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