A Hamilton-Jacobi Formulation for Optimal Coordination of Heterogeneous Multiple Vehicle Systems

Abstract

We present a method for optimal coordination of multiple vehicle teams when multiple endpoint configurations are equally desirable, such as seen in the autonomous assembly of formation flight. The individual vehicles' positions in the formation are not assigned a priori and a key challenge is to find the optimal configuration assignment along with the optimal control and trajectory. Commonly, assignment and trajectory planning problems are solved separately. We introduce a new multi-vehicle coordination paradigm, where the optimal goal assignment and optimal vehicle trajectories are found simultaneously from a viscosity solution of a single Hamilton-Jacobi (HJ) partial differential equation (PDE), which provides a necessary and sufficient condition for global optimality. Intrinsic in this approach is that individual vehicle dynamic models need not be the same, and therefore can be applied to heterogeneous systems. Numerical methods to solve the HJ equation have historically relied on a discrete grid of the solution space and exhibits exponential scaling with system dimension, preventing their applicability to multiple vehicle systems. By utilizing a generalization of the Hopf formula, we avoid the use of grids and present a method that exhibits polynomial scaling in the number of vehicles.

Reachability as a Unifying Framework for Computing Helicopter Safe Operating Conditions and Autonomous Emergency Landing

“…These techniques were used in Darbon and Osher (2016) to compute solutions to the HJ equation for systems with state and time-independent Hamiltonians of the form ẋ = f (u (t)), and Kirchner et al (2018a) expanded the classes of systems to general linear systems in high-dimensions. Additionally, these methods were successfully applied to vehicle control problems such as collaborative pursuit-evasion (Kirchner et al (2018b)) and coordination of heterogeneous groups of vehicles (Kirchner et al (2020)).…”