ANCHOR: Safe Diffusion Planning via Analytic Approximation of Doob's $h$-function

Diffusion models provide expressive, multimodal priors for trajectory planning, but enforcing hard safety constraints without distorting the learned trajectory distribution remains challenging. We formulate safe planning as conditional generation from the distribution of trajectories given collision avoidance, whose exact reverse process is characterized by a Doob $h$-transform. Since the corresponding safety posterior probability is generally intractable, we introduce ANCHOR, a training-free analytic approximation to the Doob guidance. Our method approximates the posterior using Tweedie moments and exploits signed-distance geometry for efficient computation. Under explicit regularity assumptions on the data distribution and obstacle geometry, we prove that the resulting guided diffusion is almost surely safe. On synthetic multimodal distributions, our method recovers the safety-conditioned distribution substantially more accurately than safe generative-planning baselines while avoiding boundary concentration. On Maze2D, it achieves boundary-safe trajectories with zero trap rate, while remaining computationally competitive with state-of-the-art safe diffusion and flow-matching planners.