A Hamilton–Jacobi Reachability-Based Minimum Separation Estimation of Integrated Manned and Unmanned Operation in Uncertain Environments

This work presents a minimum separation calculation for the integrated operation of manned and unmanned aerial vehicles in an uncertain airspace environment. Different from traditional path-planning-based research, this study investigated the minimum safe separation distance from a novel perspective of reachability analysis. The proposed computational method made use of the Hamilton–Jacobi partial differential equation (HJPDE) to obtain the backward reachable tube. Firstly, this work modeled the integrated operation in the UAS traffic management scenario, particularly focusing on the uncertainties. Then, a probabilistic reachability tube computation method was derived. Next, this work calculated the safe separation distances based on reachability analysis for three scenarios: a deterministic environment, an environment with relative position uncertainty, and an environment with relative heading angle uncertainty. By calculating the reachable tubes for a given response time, the worst-case minimum safe distances from the UAV’s perspective were determined, and the quantitative patterns were summarized. The results in this work indicate that, with an increase in the risk level and under the premise of a 1 second response time, the minimum safe separation increases from 26.7 meters to 30.0 meters. Finally, the paper discusses the results, explaining their rationality from both mathematical and physical perspectives.