Reachability Calculations for Vehicle Safety During Manned/Unmanned Vehicle Interaction

Abstract

This paper describes an approach based on reachability calculations for ensuring robust operation guarantees in flight maneuver sequences performed by unmanned aerial vehicles under supervision of human operators, with applications to safety-critical scenarios. Using a hybrid system formalism to model the maneuver sequence, the paper devises systematic procedures for designing switching conditions to ensure the properties of safety, target attainability, and invariance, using Hamilton-Jacobi reachability calculations. These calculations lay the foundations for refining or designing protocols for multiple unmanned aerial vehicle and/or manned vehicle interaction. The mathematical foundations necessary are described in order to formulate verification problems on reachability and safety of flight maneuvers, including issues of command latency and disturbance. An example of this formalism is given in the context of automated aerial refueling, to inform unmanned aerial vehicle decisions that avoid unsafe scenarios while achieving mission objectives. Nomenclature A = avoid set for all maneuvers A f i = collision set for mode q i , using controller K i , over time interval 0; i , A; K i ; i a 0 = unsafe radius, m B = disk in the plane D = disturbance input space d = disturbance input vector dom = domain of flight maneuvers d 1 = disturbance input, translational velocity of tanker, m=s f = vector field defining continuous dynamics H = Hamiltonian for capture and collision-set calculation K i = feedback control law designed for maneuver q i L = communication latency, s p = costate vector q i = transition maneuver q i = stationary maneuver q i = escape maneuver R f i = capture set for target R, in mode q i , using controller K i , over time interval 0; i , R; K i ; i R i = target set for maneuver q i t = time, s U = control input space u = control input vector u 1 = control input, translational velocity of unmanned aerial vehicle, m=s u 2 = control input, angular velocity of unmanned aerial vehicle, rad=s V = neighborhood of states around the fuel boom v 0 = nominal translational velocity of tanker, m=s W 0 = subset of R n used to initialize an invariant-set calculation W 1 f = invariant set under vector field f over infinite time horizon 0; 1 X 0 = set of permissible initial continuous states x = continuous state vector x 1 = longitudinal distance from the tanker to the unmanned aerial vehicle, m x 1 f = desired longitudinal distance from the tanker to the unmanned aerial vehicle, m x 2 = lateral distance from the tanker to the unmanned aerial vehicle, m x 2 f = desired lateral distance from the tanker to the unmanned aerial vehicle, m x 3= heading of the tanker relative to the unmanned aerial vehicle, rad ii1 = guard condition specifying set of states for which transition is enabled between q i and q i1 ii1 = transition command between maneuver q i and q i1 = time interval, s = solution to a Hamilton-Jacobi equation

Methods for Reachability-based Hybrid Controller Design

“…Parts of the material presented here have appeared previously in several papers: Ding et al (2008); Ding and Tomlin (2010); Ding et al (2011b,a); Kamgarpour et al (2011);Ding et al (2012). In the following, we provide an overview of the main themes from each of the subsequent chapters.…”

Towards docking safety analysis for unmanned aerial vehicle probe-drogue autonomous aerial refueling based on docking success-probability and docking reachability

“…Hamilton–Jacobi partial differential equation (H-J PDE) can be used to calculate the docking reachable set A. 31,33 The docking reachable set A and the docking target set B are equal to the zero level set of the function J(xUAV,t) at t=-τ and t=0, respectively. Namely, …”

“…The reachability calculation method has been applied to solve the following problems: collision avoidance for dynamic games, 25 reachability analysis to safe maneuvers, 26–28 vehicle reachability and safety during AAR. 29–31…”

Antagonistic control

“…It has been demonstrated how these attacks can occur and proposals have been made for how to protect against these attacks [7,8,9,10]. While the approaches above are typically concerned with monitoring and detecting attacks, there is an alternative approach which attempts to constrain the system so that a catastrophic event cannot occur [11,12]. Another branch of inquiry models the problem from a game theoretic perspective, often asking how best to invest resources to defend a system [13,14].…”