Minimal time delivery of multiple robots

Consider a set of autonomous vehicles, each one with a preassigned task to start at a given region. Due to energy constraints, and in order to minimize the overall task completion time, these vehicles are deployed from a faster carrier vehicle. This paper develops a dynamic programming (DP) based solution for the problem of finding the optimal deployment location and time for each vehicle, and for a given sequence of deployments, so that the global mission duration is minimal. The problem is specialized for ocean-going vehicles operating under time-varying currents. The solution approach involves solving a sequence of optimal stopping problems that are transformed into a set variational inequalities through the application of the dynamic programming principle (DPP). The optimal trajectory for the carrier and the optimal deployment location and time for each vehicle to be deployed are obtained in feedback-form from the numerical solution of the variational inequalities. The solution is computed with our open source parallel implementation of the fast sweeping method. The approach is illustrated with two numerical examples.