# Reachability-Based Safety Guarantees using Efficient Initializations

## TL;DR

This paper introduces a warm-start approach for Hamilton-Jacobi-Isaacs reachability analysis, significantly reducing computation time while maintaining safety guarantees, especially useful for dynamic and uncertain autonomous systems.

## Contribution

The paper proposes a novel warm-start method for reachability analysis that guarantees conservative solutions and accelerates convergence compared to standard techniques.

## Key findings

- Warm-starting results in guaranteed conservative solutions.
- The method achieves up to 6.2 times faster computation than discounted reachability.
- Demonstrated effectiveness on a 10D quadcopter model with changing conditions.

## Abstract

Hamilton-Jacobi-Isaacs (HJI) reachability analysis is a powerful tool for analyzing the safety of autonomous systems. This analysis is computationally intensive and typically performed offline. Online, however, the autonomous system may experience changes in system dynamics, external disturbances, and/or the surrounding environment, requiring updated safety guarantees. Rather than restarting the safety analysis, we propose a method of "warm-start" reachability, which uses a user-defined initialization (typically the previously computed solution). By starting with an HJI function that is closer to the solution than the standard initialization, convergence may take fewer iterations. In this paper we prove that warm-starting will result in guaranteed conservative solutions by over-approximating the states that must be avoided to maintain safety. We additionally prove that for many common problem formulations, warm-starting will result in exact solutions.We demonstrate our method on several illustrative examples with a double integrator, and also on a more practical example with a 10D quadcopter model that experiences changes in mass and disturbances and must update its safety guarantees accordingly. We compare our approach to standard reachability and a recently proposed "discounted" reachability method, and find for our examples that warm-starting is 1.6 times faster than standard and 6.2 times faster than (untuned) discounted reachability.