Showing papers by "Swaroop Darbha published in 2012"

Sampling-Based Path Planning for a Visual Reconnaissance Unmanned Air Vehicle

Abstract: Nomenclature jAj = cardinality of a set A A , A, @A = interior, closure, and boundary of a set A, respectively C = cost of an aircraft reconnaissance tour , m d x;x0 = length of shortest aircraft path from state x to state x0, m nsamples = actual number of samples to build a roadmap nsamples = estimated number of samples to build a roadmap rmin = Dubins aircraft minimum turn radius R = s-dimensional Euclidean space S = circle parameterized by angle radians ranging from 0 to 2 SE(2) = special Euclidean group R S T = set fT 1; T 2; . . . ; T ng of n targets to be photographed by aircraft V T i = visibility region of ith target Va = Dubins aircraft airspeed X = aircraft state space x = aircraft state vector x; y = Dubins aircraft Earth-fixed Cartesian coordinates, m = parameter controls ratio of translational vs rotational density of roadmap = Dubins aircraft azimuth angle, rad 2 = set of all subsets of a set A

UAV search & capture of a moving ground target under delayed information

Abstract: The optimal control of a UAV searching for a slower moving ground target on a road network is considered. To aid the UAV, the road network has been instrumented with Unattended Ground Sensors (UGSs). The target/intruder heads towards a protected goal region, oblivious of being tracked by the UAV. Whenever the intruder reaches an intersection in the road network, we assume that he randomly chooses his future course of action. The UGSs are placed at all the road junctions and are triggered when the intruder passes by. When the UAV arrives at an UGS, the UGS informs the UAV if and when the intruder passed by. The UAV does not have on-board capability to detect and identify the intruder and therefore must investigate the UGSs to learn the intruder's location. When the intruder and the UAV are at an UGS location at the same time, the UGS is triggered and this information is instantly relayed to the UAV, which snaps a picture, thereby “capturing” the intruder. Sufficient conditions for guaranteed capture of the intruder, before he reaches the goal, and the corresponding UAV control policy, that leads to capture, are provided.

3-Approximation algorithm for a two depot, heterogeneous traveling salesman problem

Abstract: We present the first approximation algorithm for a two depot, heterogeneous traveling salesman problem with an approximation ratio of 3 when the costs are symmetric and satisfy the triangle inequality.

Optimization of Perimeter Patrol Operations Using Unmanned Aerial Vehicles

Abstract: This paper addresseses the following base perimeter patrol problem: a team of unmanned aerial vehicles (UAVs) equipped with cameras and a remotely located operator cooperatively perform the task of perimeter surveillance. There arem alert stations/sites on the perimeter where a nearby breaching of the perimeter by an intruder is flagged byanunattended ground sensor (UGS).Todeterminewhether an incursionflaggedby aUGS is a false alarmora real threat, a patrolling UAV flies to the alert site to investigate the alert. The longer a UAVdwells (loiters) at an alert site, the more information it gathers; however, this also increases the delay in responding to other alerts. The decision problem for aUAV is to determine the optimal dwell time so as tomaximize the expected payoff. In this paper, patrols consisting of one and two UAVs are considered. A stochastic dynamic programming approach is employed to obtain optimal policies for the patrolling UAVs. Theoretical performance bounds from queueing systems literature have been used to benchmark the optimal controller. Also, simulation results for the optimal patrols showing the expected information gained and response time for different alert arrival rates are presented.

Algorithms for synthesizing mechanical systems with maximal natural frequencies

Abstract: We consider a simpler version of an open problem in system realization theory, which has relevance to several important problems in biomedicine, altering the dynamic response of discrete and continuous systems, connectivity of Very Large Scale Integrated circuits, as well as the co-ordination of Unmanned vehicles. The fundamental question this article tries to answer is the following one: Given all the components of a system, how do we put these components together in order to obtain a desired response? In the simplest form, this basic question arises in mechanical systems where, the objective is to connect the masses with springs in a suitable way, and in the most general form, it arises in biomedicine where one is interested in engineering and achieving a desired output by either allowing certain new interactions or disallowing some interactions to take place between the proteins, nucleic acids and other cellular components. We formulate a simpler version of this problem in one dimension (i.e., all the masses and springs are arranged along a line), where the objective is to choose a set of springs to connect the masses so that the resulting “graph” structure is as stiff as possible. The system considered corresponds to an ungrounded structure and will always admit a rigid body mode; for that reason, the smallest natural frequency is zero and we use the smallest non-zero natural frequency as a metric for stiffness of the structure and we maximize this objective. Maximizing the smallest non-zero frequency increases all the natural frequencies thereby making the system stiffer. We develop an iterative primal-dual algorithm and a cutting plane algorithm to solve the problem and provide preliminary computational results on a network up to nine masses.

Bounding procedure for stochastic dynamic programs with application to the perimeter patrol problem

Abstract: One often encounters the curse of dimensionality in the application of dynamic programming to determine optimal policies for controlled Markov chains. In this paper, we provide a method to construct sub-optimal policies along with a bound for the deviation of such a policy from the optimum via a linear programming approach. The state-space is partitioned and the optimal cost-to-go or value function is approximated by a constant over each partition. By minimizing a positive cost function defined on the partitions, one can construct an approximate value function which also happens to be an upper bound for the optimal value function of the original Markov Decision Process (MDP). As a key result, we show that this approximate value function is independent of the positive cost function (or state dependent weights; as it is referred to, in the literature) and moreover, this is the least upper bound that one can obtain; once the partitions are specified. We apply the linear programming approach to a perimeter surveillance stochastic optimal control problem; whose structure enables efficient computation of the upper bound.

Synthesizing Robust Communication Networks for Unmanned Aerial Vehicles With Resource Constraints

Abstract: In this article, we address the problem of synthesizing UAV communication networks in the presence of resource constraints. UAVs can be deployed as backbone nodes in ad-hoc networks that can be central to civilian and military applications. The cost of operation of the network depends on the resources that are used such as the total power consumption associated with the network and the number of communication links in the network. The objective of the problem is to synthesize a communication network that maximizes connectivity subject to cost of operation being within the specified resource. We choose algebraic connectivity as a measure of connectivity of the network as it is known to be a measure of robust connectivity to random node failures in the network. We pose the network synthesis problem as a mixed-integer semi-definite program and provide (1) an algorithm for computing optimal solutions using cutting plane methods, and (2) construct feasible solutions using heuristics and estimate their quality. The network synthesis problem is a NP-hard problem and there are no guarantees on the running time of the algorithm that computes an optimal solution. We provide some computational results to corroborate the performance of the proposed algorithms.Copyright © 2012 by ASME

Sub-Optimal Stationary Policies for a Class of Stochastic Optimization Problems Arising in Robotic Surveillance Applications

Abstract: This paper deals with the development of sub-optimal decision making algorithms for a collection of robots in order to aid a remotely located human operator in the task of classification of incursions across a perimeter in a surveillance application. The operator is tasked with the classification of incursion as either a nuisance or a threat. Whenever there is an incursion into the perimeter, Unattended Ground Sensors (UGS) raise an alert and the robots service the alerts by visiting the alert location and collecting evidence in the form of video and other images and transmit them to the operator. There are two competing needs for a robot: it needs to spend more time at an alert location for aiding the operator in accurate classification and it needs to service the alerts as quickly as possible so that the evidence collected is relevant. A natural problem is to determine the optimal amount of time a robot must spend servicing an alert. In this paper, we discretize the problem spatially and temporally and recast the optimization problem as follows: Is it better for a robot to spend the next time interval at the alert location in terms of maximizing the expected, discounted payoff? The payoff associated with a state is an increasing function of the time spent by a robot servicing an alert and a decreasing function of the number of unserviced alerts. This problem can be easily be cast as a Markov Decision Process (MDP). However, the number of states runs into billions even for a modest size problem. We consider Approximate Dynamic Programming via linear programming as this approach provides an upper (and lower) bound on the optimal expected discounted payoff and enables the construction of a suboptimal policy. The bounds may then be used to provide an estimate of the quality of sub-optimal policy employed. We also provide a computationally tractable way of computing the lower bound using linear programming. Finally, numerical results supporting our method are provided.Copyright © 2012 by ASME

Synthesizing robust communication networks for UAVs

Abstract: This article addresses the problem of synthesizing communication networks with maximum algebraic connectivity in the presence of constraints which limit the total number of communication links present in the network. This problem arises in Unmanned Aerial Vehicle (UAV) monitoring applications where some UAVs have to be deployed to relay and transmit time sensitive information between all the vehicles and the control stations. This network synthesis problem is a difficult optimization problem because of its non-linear objective coupled with the possibility that the number of feasible solutions increases rapidly with the size of the graph. The network synthesis problem is formulated as a mixed-integer, semi-definite program, and an algorithm to find the optimal solution is developed based on cutting plane and bisection methods. Some computational results are also presented to corroborate the performance of the proposed algorithm. Two other heuristics are presented along with numerical results corroborating their performance. Since these heuristics are based on evaluating the spectrum of the graph, they can be applied to large networks.

Computation of a Lower Bound for a Vehicle Routing Problem With Motion Constraints

Abstract: Given a set of targets that need to be monitored and a vehicle, we consider a combinatorial motion planning problem where the objective is to find a path for the vehicle such that each target is visited at least once by the vehicle, the path satisfies the motion constraints of the vehicle and the length of the path is a minimum. This is an NP-hard problem and currently, there are no algorithms that can find an optimal solution to this problem. In this article, we model the motion of the vehicle as a Dubins car and develop a method that can provide tight lower bounds to the motion planning problem. We accomplish this by relaxing the constraints corresponding to the angle of approach at each of the targets and then penalizing them whenever they are violated. The solution to the Lagrangian relaxation gives a lower bound, and this lower bound is maximized over the penalty variables using subgradient optimization. The proposed method is the first of its kind for finding tight lower bounds for combinatorial motion planning problems and can be extended to similar problems with more general motion constraints.Copyright © 2012 by ASME

On Controlling the Transient Response of Linear Time Invariant Systems with Fixed Structure Controllers

Abstract: The problem of controlling transient response is important in many industrial applications; for example, the speed and accuracy of motion control of robots directly relates to the productivity of the robot. The objective of transient control is to determine a feedback controller of a fixed structure that renders the closed loop response of a specified system to lie in a specified envelope. One may associate a set of errors which measures the deviation of the response from the envelope. The set of errors may be defined in such a way that all the errors are non-negative if and only if the response does not deviate from the envelope at any time. The transient problem can be thus posed as the problem of determining a stabilizing controller that renders the set of all errors to be non-negative at every time. One may associate a control parameter vector K with a controller of a specified structure. The main topic of investigation of this paper is to find a bound for the set of real control parameters, K, so that a rational, proper transfer function, has a decaying, non-negative impulse response. It is assumed that the coefficients of the polynomials N(s, k) and D(s, K) are affine in K.

Algorithms for Finding Diameter-constrained Graphs with Maximum Algebraic Connectivity

Abstract: This article addresses a problem of synthesizing robust networks in the presence of constraints which limit the maximum number of links that connect any two nodes in the network. This problem arises in surveillance and monitoring applications where wireless sensor networks have to be deployed to collect and exchange time sensitive information among the vehicles. This network synthesis problem is formulated as a mixed-integer, semi-definite program, and an algorithm for finding the optimal solution is developed based on cutting plane and bisection methods. Computational results are presented to corroborate the performance of the proposed algorithm.

Output Regulation of a Class of Nonlinear Systems Using Differential-Algebraic Equations

Abstract: This paper deals with the problem of synthesizing feedforward control to aid the regulation of output of a nonlinear system in the presence of partially known exogenous inputs. Currently known methods for this problem either require the solution of a constrained partial differential equation or the preview information of the signal to be tracked. The novelty of this paper lies in synthesizing feedforward control as the solution of an algebraic - differential equation, which is considerably less complex. The techniques developed in this paper generalize very directly to nonlinear systems governed by differential-algebraic equations. In this paper, we consider two separate problems: the problem of tracking reference signals and the problem of regulating the output while rejecting the disturbances. We assume that the disturbance and the reference signals are outputs of known exogenous systems. Furthermore, we assume that the initial conditions for the exogenous system corresponding to the reference signals are known while those for the exogenous system corresponding to disturbances are unknown. We develop a parameter identification scheme to estimate the unknown initial conditions for the exogenous system in the case of output regulation. We illustrate the effectiveness of the control schemes for tracking problem with the example of a ball and beam system, and for the disturbance attenuation problem with two examples, nonlinear vibration absorber and a rotational translational actuator.Copyright © 2012 by ASME

Algorithms for routing vehicles and their application to the paratransit vehicle scheduling problem.

Abstract: "As the demand for paratransit services increases, there is a constant pressure to maintain the quality of service provided to the customers while minimizing the cost of operation: this is especially important as the availability of public funding for paratransit services has been on the decline. Key tasks in accomplishing this objective are efficiently allocating vehicles to service trips and adjusting the schedules of vehicles dynamically in response to calls received by the service providers from the customers on the day of the service."

A Lower Bounding Linear Programming approach to the Perimeter Patrol Stochastic Control Problem.

Abstract: One encounters the curse of dimensionality in the application of dynamic programming to determine optimal policies for large scale controlled Markov chains. In this article, we consider a perimeter patrol stochastic optimal control problem. To determine the optimal control policy, one has to solve a Markov decision problem, whose large size renders exact dynamic programming methods intractable. So, we propose a state aggregation based approximate linear programming method to construct provably good sub-optimal policies instead. The state-space is partitioned and the optimal cost-to-go or value function is restricted to be a constant over each partition. We show that the resulting restricted system of linear inequalities embeds a family of Markov chains of lower dimension, one of which can be used to construct a tight lower bound on the optimal value function. In general, the construction of the lower bound requires the solution to a combinatorial problem. But the perimeter patrol problem exhibits a special structure that enables tractable linear programming formulation for the lower bound. We demonstrate this and also provide numerical results that corroborate the efficacy of the proposed methodology.