A Complete Algorithm for a Moving Target Traveling Salesman Problem with Obstacles

The moving target traveling salesman problem with obstacles (MT-TSP-O) is a generalization of the traveling salesman problem (TSP) where, as its name suggests, the targets are moving. A solution to the MT-TSP-O is a trajectory that visits each moving target during a certain time window(s), and this trajectory avoids stationary obstacles. We assume each target moves at a constant velocity during each of its time windows. The agent has a speed limit, and this speed limit is no smaller than any target's speed. This paper presents the first complete algorithm for finding feasible solutions to the MT-TSP-O. Our algorithm builds a tree where the nodes are agent trajectories intercepting a unique sequence of targets within a unique sequence of time windows. We generate each of a parent node's children by extending the parent's trajectory to intercept one additional target, each child corresponding to a different choice of target and time window. This extension consists of planning a trajectory from the parent trajectory's final point in space-time to a moving target. To solve this point-to-moving-target subproblem, we define a novel generalization of a visibility graph called a moving target visibility graph (MTVG). Our overall algorithm is called MTVG-TSP. To validate MTVG-TSP, we test it on 570 instances with up to 30 targets. We implement a baseline method that samples trajectories of targets into points, based on prior work on special cases of the MT-TSP-O. MTVG-TSP finds feasible solutions in all cases where the baseline does, and when the sum of the targets' time window lengths enters a critical range, MTVG-TSP finds a feasible solution with up to 38 times less computation time.

The Persistent Robot Charging Problem for Long-Duration Autonomy

This paper introduces a novel formulation aimed at determining the optimal schedule for recharging a fleet of $n$ heterogeneous robots, with the primary objective of minimizing resource utilization. This study provides a foundational framework applicable to Multi-Robot Mission Planning, particularly in scenarios demanding Long-Duration Autonomy (LDA) or other contexts that necessitate periodic recharging of multiple robots. A novel Integer Linear Programming (ILP) model is proposed to calculate the optimal initial conditions (partial charge) for individual robots, leading to the minimal utilization of charging stations. This formulation was further generalized to maximize the servicing time for robots given adequate charging stations. The efficacy of the proposed formulation is evaluated through a comparative analysis, measuring its performance against the thrift price scheduling algorithm documented in the existing literature. The findings not only validate the effectiveness of the proposed approach but also underscore its potential as a valuable tool in optimizing resource allocation for a range of robotic and engineering applications.

$A^*$ for Graphs of Convex Sets

We present a novel algorithm that fuses the existing convex-programming based approach with heuristic information to find optimality guarantees and near-optimal paths for the Shortest Path Problem in the Graph of Convex Sets (SPP-GCS). Our method, inspired by $A^*$, initiates a best-first-like procedure from a designated subset of vertices and iteratively expands it until further growth is neither possible nor beneficial. Traditionally, obtaining solutions with bounds for an optimization problem involves solving a relaxation, modifying the relaxed solution to a feasible one, and then comparing the two solutions to establish bounds. However, for SPP-GCS, we demonstrate that reversing this process can be more advantageous, especially with Euclidean travel costs. In other words, we initially employ $A^*$ to find a feasible solution for SPP-GCS, then solve a convex relaxation restricted to the vertices explored by $A^*$ to obtain a relaxed solution, and finally, compare the solutions to derive bounds. We present numerical results to highlight the advantages of our algorithm over the existing approach in terms of the sizes of the convex programs solved and computation time.

A Mixed-Integer Conic Program for the Moving-Target Traveling Salesman Problem based on a Graph of Convex Sets

This paper introduces a new formulation that finds the optimum for the Moving-Target Traveling Salesman Problem (MT-TSP), which seeks to find a shortest path for an agent, that starts at a depot, visits a set of moving targets exactly once within their assigned time-windows, and returns to the depot. The formulation relies on the key idea that when the targets move along lines, their trajectories become convex sets within the space-time coordinate system. The problem then reduces to finding the shortest path within a graph of convex sets, subject to some speed constraints. We compare our formulation with the current state-of-the-art Mixed Integer Conic Program (MICP) solver for the MT-TSP. The experimental results show that our formulation outperforms the MICP for instances with up to 20 targets, with up to two orders of magnitude reduction in runtime, and up to a 60\% tighter optimality gap. We also show that the solution cost from the convex relaxation of our formulation provides significantly tighter lower bounds for the MT-TSP than the ones from the MICP.

Assisted Path Planning for a UGV-UAV Team Through a Stochastic Network

In this article, we consider a multi-agent path planning problem in a stochastic environment. The environment, which can be an urban road network, is represented by a graph where the travel time for selected road segments (impeded edges) is a random variable because of traffic congestion. An unmanned ground vehicle (UGV) wishes to travel from a starting location to a destination while minimizing the arrival time at the destination. UGV can traverse through an impeded edge but the true travel time is only realized at the end of that edge. This implies that the UGV can potentially get stuck in an impeded edge with high travel time. A support vehicle, such as an unmanned aerial vehicle (UAV) is simultaneously deployed from its starting position to assist the UGV by inspecting and realizing the true cost of impeded edges. With the updated information from UAV, UGV can efficiently reroute its path to the destination. The UGV does not wait at any time until it reaches the destination. The UAV is permitted to terminate its path at any vertex. The goal is then to develop an online algorithm to determine efficient paths for the UGV and the UAV based on the current information so that the UGV reaches the destination in minimum time. We refer to this problem as Stochastic Assisted Path Planning (SAPP). We present Dynamic $k$-Shortest Path Planning (D*KSPP) algorithm for the UGV planning and Rural Postman Problem (RPP) formulation for the UAV planning. Due to the scalability challenges of RPP, we also present a heuristic based Priority Assignment Algorithm (PAA) for the UAV planning. Computational results are presented to corroborate the effectiveness of the proposed algorithm to solve SAPP.

DMS*: Minimizing Makespan for Multi-Agent Combinatorial Path Finding

Multi-Agent Combinatorial Path Finding (MCPF) seeks collision-free paths for multiple agents from their initial to goal locations, while visiting a set of intermediate target locations in the middle of the paths. MCPF is challenging as it involves both planning collision-free paths for multiple agents and target sequencing, i.e., solving traveling salesman problems to assign targets to and find the visiting order for the agents. Recent work develops methods to address MCPF while minimizing the sum of individual arrival times at goals. Such a problem formulation may result in paths with different arrival times and lead to a long makespan, the maximum arrival time, among the agents. This paper proposes a min-max variant of MCPF, denoted as MCPF-max, that minimizes the makespan of the agents. While the existing methods (such as MS*) for MCPF can be adapted to solve MCPF-max, we further develop two new techniques based on MS* to defer the expensive target sequencing during planning to expedite the overall computation. We analyze the properties of the resulting algorithm Deferred MS* (DMS*), and test DMS* with up to 20 agents and 80 targets. We demonstrate the use of DMS* on differential-drive robots.

C*: A New Bounding Approach for the Moving-Target Traveling Salesman Problem

We introduce a new bounding approach called Continuity* (C*) that provides optimality guarantees to the Moving-Target Traveling Salesman Problem (MT-TSP). Our approach relies on relaxing the continuity constraints on the agent's tour. This is done by partitioning the targets' trajectories into small sub-segments and allowing the agent to arrive at any point in one of the sub-segments and depart from any point in the same sub-segment when visiting each target. This lets us pose the bounding problem as a Generalized Traveling Salesman Problem (GTSP) in a graph where the cost of traveling an edge requires us to solve a new problem called the Shortest Feasible Travel (SFT). We also introduce C*-lite, which follows the same approach as C*, but uses simple and easy to compute lower-bounds to the SFT. We first prove that the proposed algorithms provide lower bounds to the MT-TSP. We also provide computational results to corroborate the performance of C* and C*-lite for instances with up to 15 targets. For the special case where targets travel along lines, we compare our C* variants with the SOCP based method, which is the current state-of-the-art solver for MT-TSP. While the SOCP based method performs well for instances with 5 and 10 targets, C* outperforms the SOCP based method for instances with 15 targets. For the general case, on average, our approaches find feasible solutions within ~4% of the lower bounds for the tested instances.

Heuristic Search for Path Finding with Refuelling

This paper considers a generalization of the Path Finding (PF) with refueling constraints referred to as the Refuelling Path Finding (RF-PF) problem. Just like PF, the RF-PF problem is defined over a graph, where vertices are gas stations with known fuel prices, and edge costs depend on the gas consumption between the corresponding vertices. RF-PF seeks a minimum-cost path from the start to the goal vertex for a robot with a limited gas tank and a limited number of refuelling stops. While RF-PF is polynomial-time solvable, it remains a challenge to quickly compute an optimal solution in practice since the robot needs to simultaneously determine the path, where to make the stops, and the amount to refuel at each stop. This paper develops a heuristic search algorithm called Refuel A* (RF-A* ) that iteratively constructs partial solution paths from the start to the goal guided by a heuristic function while leveraging dominance rules for state pruning during planning. RF-A* is guaranteed to find an optimal solution and runs more than an order of magnitude faster than the existing state of the art (a polynomial time algorithm) when tested in large city maps with hundreds of gas stations.

Enhanced Multi-Objective A* with Partial Expansion

The Multi-Objective Shortest Path Problem, typically posed on a graph, determines a set of paths from a start vertex to a destination vertex while optimizing multiple objectives. In general, there does not exist a single solution path that can simultaneously optimize all the objectives and the problem thus seeks to find a set of so-called Pareto-optimal solutions. To address this problem, several Multi-Objective A* (MOA*) algorithms were recently developed to quickly compute solutions with quality guarantees. However, these MOA* algorithms often suffer from high memory usage, especially when the branching factor (i.e., the number of neighbors of any vertex) of the graph is large. This work thus aims at reducing the high memory consumption of MOA* with little increase in the runtime. In this paper, we first extend the notion of "partial expansion" (PE) from single-objective to multi-objective and then fuse this new PE technique with EMOA*, a recent runtime efficient MOA* algorithm. Furthermore, the resulting algorithm PE-EMOA* can balance between runtime and memory efficiency by tuning a user-defined hyper-parameter.

Cooperative Coverage with a Leader and a Wingmate in Communication-Constrained Environments

We consider a mission framework in which two unmanned vehicles (UVs), a leader and a wingmate, are required to provide cooperative coverage of an environment while being within a short communication range. This framework finds applications in underwater and/or military domains, where certain constraints are imposed on communication by either the application or the environment. An important objective of missions within this framework is to minimize the total travel and communication costs of the leader-wingmate duo. In this paper, we propose and formulate the problem of finding routes for the UVs that minimize the sum of their travel and communication costs as a network optimization problem of the form of a binary program (BP). The BP is computationally expensive, with the time required to compute optimal solutions increasing rapidly with the problem size. To address this challenge, here, we propose two algorithms, an approximation algorithm and a heuristic algorithm, to solve large-scale instances of the problem swiftly. We demonstrate the effectiveness and the scalability of these algorithms through an analysis of extensive numerical simulations performed over 500 instances, with the number of targets in the instances ranging from 6 to 100.