$λ$-Reachability: Geometric-Horizon Safety Bellman Equations for Humanoid Safety

2026-06-14

AI Summary

This work addresses the challenge of ensuring long-term safety while maintaining computational efficiency in high-dimensional humanoid robot systems. The authors propose λ-Reachability, a novel method that constructs a multi-step safety value estimator by combining geometrically distributed rollout horizons with stochastic absorbing terminals, thereby interpolating between locally self-consistent updates and long-horizon maximal trajectory safety objectives. A tunable parameter δ controls the probability of utilizing terminal safety values, and the approach establishes, for the first time, a theoretical connection between λ and undiscounted reachability objectives—offering both contraction mapping properties and long-horizon safety estimation capabilities. Integrated with Hamilton-Jacobi safety analysis and a TD(λ)-like learning mechanism, the method significantly outperforms single-step baselines in simulated and real-world humanoid balancing and obstacle avoidance tasks, demonstrating marked improvements in identifying safe set boundaries and estimating safety margins.

Abstract

We introduce $λ$-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, $λ$-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($λ$), $λ$-Reachability interpolates between local self-consistency updates and long-horizon max-over-trajectory safety targets via an interpretable horizon-control parameter. Unlike TD($λ$), where the terminal value is always incorporated in learning targets, the terminal safety value in $λ$-Reachability is only used at a probability controlled by parameter $δ$. We formally show that for $δ<1$, the update induces a contraction mapping that allows temporal-difference learning; as $λ\to 1$, the estimator recovers the undiscounted reachability objective. We apply $λ$-Reachability to high-dimensional safety learning problems with both simulated and real humanoid robots under balance and collision avoidance constraints. Experimental results demonstrate that $λ$-Reachability significantly improves both safe-set boundary classification and safety margin estimation compared to single-step temporal-difference baselines.

Problem

• safety analysis
• high-dimensional robotic systems
• reachability
• humanoid robots
• safe-set estimation

Innovation

• λ-Reachability
• Hamilton-Jacobi safety analysis
• multi-step safety estimator
• geometric-horizon Bellman equation
• humanoid robot safety