Ketan Savla

Traveling Salesperson Problems for the Dubins Vehicle

In this paper, we study minimum-time motion planning and routing problems for the Dubins vehicle, i.e., a nonholonomic vehicle that is constrained to move along planar paths of bounded curvature, without reversing direction. Motivated by autonomous aerial vehicle applications, we consider the traveling salesperson problem for the Dubins vehicle (DTSP): given n points on a plane, what is the shortest Dubins tour through these points, and what is its length? First, we show that the worst-case length of such a tour grows linearly with n and we propose a novel algorithm with performance within a constant factor of the optimum for the worst-case point sets. In doing this, we also obtain an upper bound on the optimal length in the classical point-to-point problem. Second, we study a stochastic version of the DTSP where the n targets are randomly and independently sampled from a uniform distribution. We show that the expected length of such a tour is of order at least n 2/3 and we propose a novel algorithm yielding a solution with length of order n 2/3 with probability one. Third and finally, we study a dynamic version of the DTSP: given a stochastic process that generates target points, is there a policy that guarantees that the number of unvisited points does not diverge over time? If such stable policies exist, what is the minimum expected time that a newly generated target waits before being visited by the vehicle? We propose a novel stabilizing algorithm such that the expected wait time is provably within a constant factor from the optimum.

Dynamic Vehicle Routing for Robotic Systems

Recent years have witnessed great advancements in the science and technology of autonomy, robotics, and networking. This paper surveys recent concepts and algorithms for dynamic vehicle routing (DVR), that is, for the automatic planning of optimal multivehicle routes to perform tasks that are generated over time by an exogenous process. We consider a rich variety of scenarios relevant for robotic applications. We begin by reviewing the basic DVR problem: demands for service arrive at random locations at random times and a vehicle travels to provide on-site service while minimizing the expected wait time of the demands. Next, we treat different multivehicle scenarios based on different models for demands (e.g., demands with different priority levels and impatient demands), vehicles (e.g., motion constraints, communication, and sensing capabilities), and tasks. The performance criterion used in these scenarios is either the expected wait time of the demands or the fraction of demands serviced successfully. In each specific DVR scenario, we adopt a rigorous technical approach that relies upon methods from queueing theory, combinatorial optimization, and stochastic geometry. First, we establish fundamental limits on the achievable performance, including limits on stability and quality of service. Second, we design algorithms, and provide provable guarantees on their performance with respect to the fundamental limits.

Efficient Routing Algorithms for Multiple Vehicles With no Explicit Communications

In this paper, we consider a class of dynamic vehicle routing problems, in which a number of mobile agents in the plane must visit target points generated over time by a stochastic process. It is desired to design motion coordination strategies in order to minimize the expected time between the appearance of a target point and the time it is visited by one of the agents. We propose control strategies that, while making minimal or no assumptions on communications between agents, provide the same level of steady-state performance achieved by the best known decentralized strategies. In other words, we demonstrate that inter-agent communication does not improve the efficiency of such systems, but merely affects the rate of convergence to the steady state. Furthermore, the proposed strategies do not rely on the knowledge of the details of the underlying stochastic process. Finally, we show that our proposed strategies yield an efficient, pure Nash equilibrium in a game theoretic formulation of the problem, in which each agents objective is to maximize the expected value of the ldquotime spent alonerdquo at the next target location. Simulation results are presented and discussed.

Robust Distributed Routing in Dynamical Networks–Part II: Strong Resilience, Equilibrium Selection and Cascaded Failures

Strong resilience properties of dynamical networks are analyzed for distributed routing policies. The latter are characterized by the property that the way the outflow at a non-destination node gets split among its outgoing links is allowed to depend only on local information about the current particle densities on the outgoing links. The strong resilience of the network is defined as the infimum sum of link-wise flow capacity reductions making the asymptotic total inflow to the destination node strictly less than the total outflow at the origin. A class of distributed routing policies that are responsive to local information is shown to yield the maximum possible strong resilience under such local information constraints for an acyclic dynamical network with a single origin-destination pair. The maximal achievable strong resilience is shown to be equal to the minimum node residual capacity of the network. The latter depends on the limit flow of the unperturbed network and is defined as the minimum, among all the non-destination nodes, of the sum, over all the links outgoing from the node, of the differences between the maximum flow capacity and the limit flow of the unperturbed network. We propose a simple convex optimization problem to solve for equilibrium flows of the unperturbed network that minimize average delay subject to strong resilience guarantees, and discuss the use of tolls to induce such an equilibrium flow in traffic networks. Finally, we present illustrative simulations to discuss the connection between cascaded failures and the resilience properties of the network.

Robust Distributed Routing in Dynamical Networks—Part I: Locally Responsive Policies and Weak Resilience

Robustness of distributed routing policies is studied for dynamical networks, with respect to adversarial disturbances that reduce the link flow capacities. A dynamical network is modeled as a system of ordinary differential equations derived from mass conservation laws on a directed acyclic graph with a single origin-destination pair and a constant total outflow at the origin. Routing policies regulate the way the total outflow at each nondestination node gets split among its outgoing links as a function of the current particle density, while the outflow of a link is modeled to depend on the current particle density on that link through a flow function. The dynamical network is called partially transferring if the total inflow at the destination node is asymptotically bounded away from zero, and its weak resilience is measured as the minimum sum of the link-wise magnitude of disturbances that make it not partially transferring. The weak resilience of a dynamical network with arbitrary routing policy is shown to be upper bounded by the networks min-cut capacity and, hence, is independent of the initial flow conditions. Moreover, a class of distributed routing policies that rely exclusively on local information on the particle densities, and are locally responsive to that, is shown to yield such maximal weak resilience. These results imply that locality constraints on the information available to the routing policies do not cause loss of weak resilience. Fundamental properties of dynamical networks driven by locally responsive distributed routing policies are analyzed in detail, including global convergence to a unique limit flow. The derivation of these properties exploits the cooperative nature of these dynamical systems, together with an additional stability property implied by the assumption of monotonicity of the flow as a function of the density on each link.

On Multiple UAV Routing with Stochastic Targets: Performance Bounds and Algorithms

In this paper we consider the following problem. A number of Uninhabited Aerial Vehicles (UAVs), modeled as vehicles moving at constant speed along paths of bounded curvature, must visit stochastically-generated targets in a convex, compact region of the plane. Targets are generated according to a spatio-temporal Poisson process, uniformly in the region. It is desired to minimize the expected waiting time between the appearance of a target, and the time it is visited. We present partially centralized algorithms for UAV routing, assigning regions of responsibility to each vehicle, and compare their performance with respect to asymptotic performance bounds, in the light and heavy load limits. Simulation results are presented and discussed.

Human-in-the-loop vehicle routing policies for dynamic environments

In this paper we design coordination policies for a routing problem requiring human-assisted classification of targets through analysis of information gathered on-site by autonomous vehicles. More precisely, we consider the following problem: targets are generated according to a spatio-temporal Poisson process, uniformly in a region of interest. It is desired to classify targets as friends or foes. In order to enable human operators to classify a target, one of the vehicles needs to travel to the target¿s location and gather sufficient information. In other words, the autonomous vehicles provide access to on-site information, and the human operator provide the judgment capabilities necessary to process such information. The objective of our analysis is to design joint motion coordination and operator scheduling policies that minimize the expected time needed to classify a target after its appearance. In addition, we analyze how the achievable system performance depends on the number of autonomous vehicles and of human operators. We present novel coordination policies between the vehicles and operators and compare the performance of these policies with respect to asymptotic performance bounds.

A Dynamical Queue Approach to Intelligent Task Management for Human Operators

Formal methods for task management for human operators are gathering increasing attention to improve efficiency of human-in-the-loop systems. In this paper, we consider a novel dynamical queue approach to intelligent task management for human operators. We consider a model of a dynamical queue, where the service time depends on the server utilization history. The proposed queueing model is motivated by, but not restricted to, widely accepted empirical laws describing human performance as a function of mental arousal. The focus of the paper is to characterize the throughput of the dynamical queue and design corresponding maximally stabilizing task release control policies, assuming deterministic arrivals. We focus extensively on threshold policies that release a task to the server only when the server state is less than a certain threshold. When every task brings in the same deterministic amount of work, we give an exact characterization of the throughput and show that an appropriate threshold policy is maximally stabilizing. The technical approach exploits the optimality of the one-task equilibria class associated with the server dynamics. When the amount of work associated with the tasks is an independent identically distributed (i.i.d.) random variable with finite support, we show that the maximum throughput increases in comparison to the case where the tasks have the same deterministic amount of work. Finally, we provide preliminary empirical evidence in support of the applicability of the proposed approach to systems with human operators.

Throughput Optimality and Overload Behavior of Dynamical Flow Networks Under Monotone Distributed Routing

This paper investigates the throughput behavior of single-commodity dynamical flow networks governed by monotone distributed routing policies. The networks are modeled as systems of ordinary differential equations based on mass conversation laws on directed graphs with limited flow capacities on the links and constant external inflows at certain origin nodes. Under monotonicity assumptions on the routing policies, it is proven that, if the external inflow at the origin nodes does not violate any cut capacity constraints, then there exists a globally asymptotically stable equilibrium, and the network achieves maximal throughput. On the contrary, should such a constraint be violated, the network overload behavior is characterized. In particular, it is established that there exists a cut with respect to which the flow densities on every link grow linearly over time (respectively, reach their respective limits simultaneously) in the case where the buffer capacities are infinite (respectively, finite).

Sharing the load

In this article, we discussed the use of various spatial tessellations to determine, in the framework of partitioning policies, optimal workload share in a mobile robotic network. We also proposed efficient and spatially distributed algorithms for achieving some of these tessellations with minimum or no communication between the agents. Because of space limitations, we have not reported results of numerical experiments in this article but provided bibliographic references to publications containing such results and further details. It is interesting to note that these tessellations appear while considering different variations of the same basic problem (DTRP). It is then natural to investigate the existence of a single objective function, whose optima correspond to the various tessellations under these different variations. The game theory approach seems to be a promising one.