Time-Optimal Multi-Waypoint Mission Planning in Dynamic Environments David L. Ferris, Deepak N. Subramani, Chinmay S. Kulkarni, Patrick J. Haley Jr., and Pierre F. J. Lermusiaux * Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139 *Email: pierrel@mit.edu Abstract—The present paper demonstrates the use of exact equations to predict time-optimal mission plans for a marine vehicle that visits a number of locations in a given dynamic ocean current field. This problem bears close resemblance to that of the classic “traveling salesman”, albeit with the added complexity that the vehicle experiences a dynamic flow field while traversing the paths. The paths, or “legs”, between all goal waypoints are generated by numerically solving the exact time- optimal path planning level-set differential equations. Overall, the planning proceeds in four steps. First, current forecasts for the planning horizon is obtained utilizing our data-driven 4- D primitive equation ocean modeling system (Multidisciplinary Simulation Estimation and Assimilation System; MSEAS), forced by high-resolution tidal and real-time atmopsheric forcing fields. Second, all tour permutations are enumerated and the minimum number of times the time-optimal PDEs are to be solved is established. Third, due to the spatial and temporal dynamics, a varying start time results in different paths and durations for each leg and requires all permutations of travel to be calculated. To do so, the minimum required time-optimal PDEs are solved and the optimal travel time is computed for each leg of all enu- merated tours. Finally, the tour permutation for which travel time is minimized is identified and the corresponding time-optimal paths are computed by solving the backtracking equation. Even though the method is very efficient and the optimal path can be computed serially in real-time for common naval operations, for additional computational speed, a high-performance computing cluster was used to solve the level set calculations in parallel. Our equation and software is applied to simulations of realistic naval applications in the complex Philippines Archipelago region. Our method calculates the global optimum and can serve two purposes: (a) it can be used in its present form to plan multi- waypoint missions offline in conjunction with a predictive ocean current modeling system, or (b) it can be used as a litmus test for approximate future solutions to the traveling salesman problem in dynamic flow fields. I. I NTRODUCTION Autonomous underwater vehicles (AUVs) are currently fielded world-wide by commercial companies, militaries, and research institutions, and their use is only set to increase in the coming years (e.g., [1]). For example, signifying the navy’s emphasize on unmanned marine systems, the United States Navy has publicly released its Unmanned Underwater Vehicle (UUV) Master Plan and it’s AUV Requirements for 2025 [2], [3]. Among the various requirements, “multi-waypoint missions” is a growing area of emphasis from a navigational standpoint for naval operations [4]. In these missions, an AUV visits multiple target locations during a single mission. For such missions, optimally utilizing ocean flow forecasts for navigation can significantly reduce operational costs. Recent focus of AUV path planning has been on com- puting exact optimal paths between starting locations and targets in strong and dynamic environments. For this purpose, we developed partial differential equations (PDEs), efficient numerical schemes, and computational systems to compute exact time-optimal paths [5] and energy-optimal paths [6] in strong and dynamic deterministic currents. We also developed stochastic PDEs to compute stochastic time-optimal paths [7] and risk-optimal paths [8] in uncertain ocean currents. We have demonstrated our path planning not only in realistic ocean re- analysis [9], [10], [11], but also in real-time with real AUVs and gliders [12], [13], [14]. Additionally, the theory, schemes and software were also extended for three-dimensional AUV path planning in realistic domains [15] and for optimal ship routing [16], [17]. In the present paper, the goal is to use our planning PDEs to predict time-optimal mission plans for a marine vehicle that visits multiple locations in a dynamic ocean flow field predicted by a data-assimilative ocean modeling system. These missions begin and end in the same location and visit a finite number of waypoints in the minimal time; this problem bears close resemblance to that of the classic “traveling salesman”, albeit with the added complexity of a dynamic flow field. Our interest is in finding an exact solution that can serve as a litmus test for future algorithmic solutions. Previous Progress. Traditionally, the focus of path planning has been on robot motion planning in static environments (e.g., [18], [19]) and recently these have been extended for AUV path planning in dynamic environments. For example, graph search schemes such as modified Dijkstra’s algorithm [20], A * search [21], and Rapidly-exploring Random Trees [22] have been used with realistic ocean flows. Other methods such as evolutionary algorithms [23], nonlinear optimization [24], wavefront expansions [25], fast marching methods [26], and LCS-based methods [27] have also been employed for AUV path planning. However, many of these methods are either inexact or computationally expensive in dynamic en- vironments. On the other hand, our PDE-based planning is exact and computationally efficient for strong, dynamic and uncertain flows. We refer the readers to [28], [14] for detailed reviews. Even though the classic traveling salesman problem and vehicle routing problems are well studied in the field of operations research (e.g., [29], [30]), the literature for Multi Waypoint AUV mission planning has been limited. In ref. [31], 978-1-5386-4814-8/18/$31.00 ©2018 IEEE