Robotic systems navigating in real-world settings require a semantic understanding of their environment to properly determine safe actions. This work aims to develop the mathematical underpinnings of such a representation -- specifically, the goal is to develop safety filters that are risk-aware. To this end, we take a two step approach: encoding an understanding of the environment via Poisson's equation, and associated risk via Laplace guidance fields. That is, we first solve a Dirichlet problem for Poisson's equation to generate a safety function that encodes system safety as its 0-superlevel set. We then separately solve a Dirichlet problem for Laplace's equation to synthesize a safe \textit{guidance field} that encodes variable levels of caution around obstacles -- by enforcing a tunable flux boundary condition. The safety function and guidance fields are then combined to define a safety constraint and used to synthesize a risk-aware safety filter which, given a semantic understanding of an environment with associated risk levels of environmental features, guarantees safety while prioritizing avoidance of higher risk obstacles. We demonstrate this method in simulation and discuss how \textit{a priori} understandings of obstacle risk can be directly incorporated into the safety filter to generate safe behaviors that are risk-aware.

We introduce a general, scalable computational framework for multi-axis 3D printing based on implicit neural fields (INFs) that unifies all stages of toolpath generation and global collision-free motion planning. In our pipeline, input models are represented as signed distance fields, with fabrication objectives such as support-free printing, surface finish quality, and extrusion control being directly encoded in the optimization of an implicit guidance field. This unified approach enables toolpath optimization across both surface and interior domains, allowing shell and infill paths to be generated via implicit field interpolation. The printing sequence and multi-axis motion are then jointly optimized over a continuous quaternion field. Our continuous formulation constructs the evolving printing object as a time-varying SDF, supporting differentiable global collision handling throughout INF-based motion planning. Compared to explicit-representation-based methods, INF-3DP achieves up to two orders of magnitude speedup and significantly reduces waypoint-to-surface error. We validate our framework on diverse, complex models and demonstrate its efficiency with physical fabrication experiments using a robot-assisted multi-axis system.