Showing papers by "Swaroop Darbha published in 2007"

A Resource Allocation Algorithm for Multivehicle Systems With Nonholonomic Constraints

Abstract: This paper is about the allocation of tours of m targets to n vehicles. The motion of the vehicles satisfies a nonholonomic constraint (i.e., the yaw rate of the vehicle is bounded). Each target is to be visited by one and only one vehicle. Given a set of targets and the yaw rate constraints on the vehicles, the problem addressed in this paper is 1) to assign each vehicle a sequence of targets to visit, and 2) to find a feasible path for each vehicle that passes through the assigned targets with a requirement that the vehicle returns to its initial position. The heading angle at each target location may not be specified. The objective function is to minimize the sum of the distances traveled by all vehicles. A constant factor approximation algorithm is presented for the above resource allocation problem for both the single and the multiple vehicle case. Note to Practitioners-The motivation for this paper stems from the need to develop resource allocation algorithms for unmanned aerial vehicles (UAVs). Small autonomous UAVs are seen as ideal platforms for many applications, such as searching for targets, mapping a given area, traffic surveillance, fire monitoring, etc. The main advantage of using these small autonomous vehicles is that they can be used in situations where a manned mission is dangerous or not possible. Resource allocation problems naturally arise in these applications where one would want to optimally assign a given set of vehicles to the tasks at hand. The feature that differentiates these resource allocation problems from similar problems previously studied in the literature is that there are constraints on the motion of the vehicle. This paper addresses the constraint that captures the inability of a fixed wing aircraft to turn at any arbitrary yaw rate. The basic problem addressed in this paper is as follows: Given n vehicles and m targets, find a path for each vehicle satisfying yaw rate contraints such that each target is visited exactly once by a vehicle and the total distance traveled by all vehicles is minimized. We assume that the targets are at least 2r apart, where r is the minimum turning radius of the vehicle. This is a reasonable assumption because the sensors on these vehicles can map or see an area whose width is at least 2r. We give an algorithm to solve this problem by combining ideas from the traveling salesman problem and the path planning literature. We also show how these algorithms perform in the worst-case scenario

A Lagrangian-Based Algorithm for a Multiple Depot, Multiple Travelling Salesmen Problem

Abstract: In this manuscript, we consider the problem of motion planning of m Dubins' vehicles through n points in a plane. The initial location and heading of the vehicles is specified and is assumed to be distinct for each vehicle. A motion plan for a vehicle is given by the sequence of points and the corresponding angles at which each point must be visited by the vehicle. We require that each vehicle return to the same initial location (depot) at the same heading after visiting the points. The objective of the motion planning problem is to choose at most q(les m) Dubins' vehicles and find their motion plans so that all the points are visited and the total cost of the tours of the chosen vehicles is a minimum amongst all the possible choice of vehicles and their tours. This problem is a generalization of the multiple depot, multiple travelling salesmen problem (MDMTSP) in two directions - the problem involves the determination of choice of vehicles and their respective headings at each of their assigned points. This problem is NP-hard. We propose a two step approach to solving this problem - (1) the combinatorial problem of choosing the vehicles and their associated tours is based on Euclidean distances between points and (2) once the sequence of points to be visited, the heading at each point is determined based on a dynamic programming scheme. The solution to the first step is based on a generalization of Held-Karp's method for the MDMTSP. We modify the Lagrangian heuristics, in the literature, for finding a close primal solution from the Held-Karp's (dual) solution. Empirical results seem to indicate that this scheme seems to provide primal solutions which are within 5% of the optimum in the span of 25 dual iterations for instances which have about 45 cities to visit and 6 vehicles.

A lagrangian-based algorithm for a combinatorial motion planning problem

Abstract: We consider a combinatorial motion planning problem (CMP) that naturally arises in many applications involving unmanned aerial vehicles (UAVs) with fuel and motion constraints. The motion constraint we consider is the inability of a vehicle to turn at an arbitrary yaw rate. The CMP is a generalization of a single travelling salesman problem and is NP-hard. In this paper, we exploit the combinatorial structure of the problem and provide heuristics with computational results to address the same.

A lagrangian-based algorithm for a combinatorial motion planning problem

Abstract: We consider a combinatorial motion planning problem (CMP) that naturally arises in many applications involving unmanned aerial vehicles (UAVs) with fuel and motion constraints. The motion constraint we consider is the inability of a vehicle to turn at an arbitrary yaw rate. The CMP is a generalization of a single Travelling Salesman Problem and is NP-Hard. In this paper, we exploit the combinatorial structure of the problem and provide heuristics with computational results to address the same.

Path Planning for a Collection of Vehicles With Yaw Rate Constraints

Abstract: Multi-vehicle systems are naturally encountered in civil and military applications. Cooperation amongst individual “miniaturized” vehicles allows for flexibility to accomplish missions that a single large vehicle may not readily be able to accomplish. While accomplishing a mission, motion planning algorithms are required to efficiently utilize a common resource (such as the total fuel in the collection of vehicles) or to minimize a collective cost function (such as the maximum time taken by the vehicles to reach their intended destination). The objective of this chapter is to present a constant factor approximation algorithm for planning the path of each vehicle in a collection of vehicles, where the motion of each vehicle must satisfy yaw rate constraints.

A new computational approach to the synthesis of fixed order controllers

Abstract: The research described in this dissertation deals with an open problem concerning the synthesis of controllers of fixed order and structure. This problem is encountered in a variety of applications. Simply put, the problem may be put as the determination of the set, S of controller parameter vectors, K = (k 1,k2,...,kl), that render Hurwitz a family (indexed by F ) of complex polynomials of the form {P0( s,α) + i=1 l Pi(s,α) ki, α ∈ F }, where the polynomials Pj(s,α), j = 0,...,l are given data. They are specified by the plant to be controlled, the structure of the controller desired and the performance that the controllers are expected to achieve. Simple examples indicate that the set S can be non-convex and even be disconnected. While the determination of the non-emptiness of S is decidable and amenable to methods such as the quantifier elimination scheme, such methods have not been computationally tractable and more importantly, do not provide a reasonable approximation for the set of controllers. Practical applications require the construction of a set of controllers that will enable a control engineer to check the satisfaction of performance criteria that may not be mathematically well characterized. The transient performance criteria often fall into this category. From the practical viewpoint of the construction of approximations for S , this dissertation is different from earlier work in the literature on this problem. A novel feature of the proposed algorithm is the exploitation of the interlacing property of Hurwitz polynomials to provide arbitrarily tight outer and inner approximation to S . The approximation is given in terms of the union of polyhedral sets which are constructed systematically using the Hermite-Biehler theorem and the generalizations of the Descartes' rule of signs.

Sequential Inspection Using Loitering

Abstract: A set of objects of interest is to be sequentially inspected by a Micro Aerial Vehicle (MAV) equipped with a camera. Upon arriving at an object of interest, an image of the object is sent to a human operator, who, upon inspecting the image, sends his feedback to the MAV. The feedback from the operator may consist of the pose angle of the object and whether he has seen any distinguishing features of the object. Upon receiving the feedback, the MAV uses this information to decide whether it should perform a secondary inspection of the object of interest or continue to the next object. A secondary inspection has a reward (or value or information gain) that is dependent on the operator’s feedback. There is an associated cost of reinspection and it depends on the delay of the operator’s feedback. It seems reasonable to let the MAV loiter for a while near the most recently inspected object of interest so that it expends a small amount of endurance from the reserve after receiving the feedback from the operator. The objective is to increase the information and hence, the total expected reward about the set of objects of interest. Since the endurance of the MAVs is limited, the loiter time near each object of interest must be carefully determined. This paper addresses the determination of the optimal loiter time through the use of Stochastic dynamic programming. Numerical results are presented that show the optimal loiter time is a function of the maximum expected operator delay.

On the Boundedness of the Set of Stabilizing Controllers

Abstract: This paper shows that the set of rational, strictly proper stabilizing controllers for single input single output (SISO) linear time invariant (LTI) plants will form a bounded (can even be empty) set in the controller parameter space if and only if the order of the stabilizing controller can not be reduced any further; if the set of proper stabilizing controllers of order tau is not empty and the set of strictly proper controllers of order r is bounded, then tau is the minimal order of stabilization. The paper also extends this result to characterize the set of controllers that guarantee some prespecified performance specifications. In particular, it is shown here that the minimal order of a controller that guarantees specified performance is I iff (1) there is a controller of order I guaranteeing the specified performance and (2) the set of strictly proper stabilizing controllers of order I and guaranteeing the performance is bounded. Moreover, if the order of the controller is increased, the set of higher order controllers which satisfies the specified performance, will necessarily be unbounded. This characterization is provided for performance specifications, such as gain margin and robust stability, which can be posed as the simultaneous stabilization of a family of real polynomials. Other performance specifications, such as phase margin and Hinfin norm, can be reduced to the problem of determining a set of stabilizing controllers that renders a family of complex polynomials Hurwitz. The characterization of the set of controllers for the stabilization of complex polynomials is provided and is used to show the boundedness properties for the set of controllers that guarantee a given phase margin or an upper bound on the Hinfin norm.

Information Flow Requirements for the Stability of Motion of Vehicles in a Rigid Formation

Abstract: It is known in the literature on Automated Highway Systems that information flow can significantly affect the propagation of errors in spacing in a collection of vehicles. This chapter investigates this issue further for a homogeneous collection of vehicles, where in the motion of each vehicle is modeled as a point mass and is digitally controlled. The structure of the controller employed by the vehicles is as follows: \( U_i (z) = C(z)\sum olimits_{j \in S_i } {(X_i - X_j - \tfrac{{L_{ij} z}} {{z - 1}})} \), where U i(z) is the (z- transformation of) control action for the i th vehicle, X i is the position of the i th vehicle, L ij is the desired distance between the i th and the j th vehicles in the collection, C(z) is the discrete transfer function of the controller and S i is the set of vehicles that the i th vehicle can communicate with directly. This chapter further assumes that the information flow is undirected, i.e., i ∈ S j ⇔ j ∈ S i and the information flow graph is connected. We consider information flow in the collection, where each vehicle can communicate with a maximum of q(n) vehicles. We allow q(n) to vary with the size n of the collection. We first show that C(z) cannot have any zeroes at z = 1 to ensure that relative spacing is maintained in response to a reference vehicle making a maneuver where its velocity experiences a steady state offset. We then show that if the control transfer function C(z) has one or more poles located at z = 1, then the motion of the collection of vehicles will become unstable if the size of the collection is sufficiently large. These two results imply that C(1) ≠ 0 and C(1) must be well defined. We further show that if q(n)/n → 0 as n → ∞ then there is a low frequency sinusoidal disturbance of at most unit amplitude acting on each vehicle such that the maximum error in spacing response increase at least as \( \Omega \left( {\sqrt {\tfrac{{n^3 }} {{q^3 (n)}}} } \right) \). A consequence of the results presented in this chapter is that the maximum of the error in spacing and velocity of any vehicle can be made insensitive to the size of the collection only if there is at least one vehicle in the collection that communicates with at least Ω(n) other vehicles in the collection. We also show that there can be at most one vehicle that communicates with Ω(n) vehicles and that any other vehicle in the collection can only communicate with at most p vehicles, where p depends only on the chosen controller and the its sampling time.