$\lambda$-Reachability: Geometric-Horizon Safety Bellman Equations for Humanoid Safety

Abstract: We introduce $\lambda$-Reachability, a scalable approach to Hamilton--Jacobi safety analysis for high-dimensional robotic systems. Unlike prior discounted formulations that rely on fixed one-step Bellman updates, $\lambda$-Reachability employs a stochastic multi-step estimator of the safety value, using a geometrically distributed rollout horizon together with a randomly absorbed terminal. Conceptually analogous to TD($\lambda$), $\lambda$-Reachability interpolates between local self-consistency updates and long-horizon max-over-trajectory safety targets via an interpretable horizon-control parameter. Unlike TD($\lambda$), where the terminal value is always incorporated in learning targets, the terminal safety value in $\lambda$-Reachability is only used at a probability controlled by parameter $\delta$. We formally show that for $\delta<1$, the update induces a contraction mapping that allows temporal-difference learning; as $\lambda \to 1$, the estimator recovers the undiscounted reachability objective. We apply $\lambda$-Reachability to high-dimensional safety learning problems with both simulated and real humanoid robots under balance and collision avoidance constraints. Experimental results demonstrate that $\lambda$-Reachability significantly improves both safe-set boundary classification and safety margin estimation compared to single-step temporal-difference baselines.

Robotics (cs.RO)

arXiv:2606.16022 [cs.RO]

(or arXiv:2606.16022v1 [cs.RO] for this version)

https://doi.org/10.48550/arXiv.2606.16022 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.