Sai Krishna Yadlapalli

Information flow and its relation to stability of the motion of vehicles in a rigid formation

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 paper investigates this issue further for a homogeneous collection of vehicles, where in the motion of each vehicle is modeled as a point mass. The structure of the controller employed by the vehicles is as follows: U/sub i/(s)=C(s)/spl Sigma/ /sub j/spl isin/si/(X/sub i/ - X/sub j/ - L/sub ij//s) where U/sub i/(s) is the (Laplace transformation of) control action for the i/sup th/ vehicle, L/sub ij/is the position of the i/sup th/ vehicle, L/sub ij/ is the desired distance between the i/sup th/ and the j/sup th/ vehicles in the collection, C(s) is the controller transfer function and S/sub i/ is the set of vehicles that the i/sup th/ vehicle can communicate with directly. This paper further assumes that the information flow is undirected, i.e., i/spl isin/S/sub j//spl harr/j/spl isin/S/sub 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, such that q(n) may vary with the size n of the collection. We first show that C(s) cannot have any zeroes at the origin 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(s) has one or more poles located at the origin of the complex plane, 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(0)/spl ne/0 and C(0) is well defined. We further show that if q(n)/sup 3//n/sup 2//spl rarr/0 as n /spl rarr//spl infin/ then there is a low frequency sinusoidal disturbance of at most unit amplitude acting on each vehicle such that the maximum errors in spacing response increase at least as O (/spl radic/(n/sup 2/)/q(n)/sup 3/). A consequence of the results presented in this paper 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 O(n/sup 2/3/) other vehicles in the collection.

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

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.

Approximation algorithms for a heterogeneous Multiple Depot Hamiltonian Path Problem

In this article, we present the first approximation algorithm for a routing problem that is frequently encountered in the motion planning of Unmanned Vehicles (UVs). The considered problem is a variant of a Multiple Depot-Terminal Hamiltonian Path Problem and is stated as follows: There is a collection of m UVs equipped with different sensors on-board and there are n targets to be visited by them collectively. There are restrictions on the targets of the following type: (1) A target may be visited by any UV, (2) a target must be visited only by a subset of UVs (with appropriate on-board sensor) and (3) a target may not be visited by a subset of UVs (as the set of on board sensors on the UV may not be suitable for viewing the targets). The UVs are otherwise identical from the viewpoint of dynamic constraints on their motion and hence, the cost of traveling from a target A to a target B is the same for all vehicles. We will assume that triangle inequality is satisfied by the cost associated with travel, i.e., it is cheaper to travel from a target A to a target B directly than to go via an intermediate target C. The UVs may possibly start from different locations (referred to as depots) and are not required to return to the depot. While there are different objectives that can be considered for this problem, we consider the total cost of travel of all the UVs as an objective to be minimized. The problem considered in this article is a generalized version of single depot-terminal Hamiltonian Path Problem and is NP-hard.

An approximation algorithm for a 2-Depot, heterogeneous vehicle routing problem

Routing problems involving heterogeneous vehicles naturally arise in several civil and military applications due to fuel and motion constraints of the vehicles. These vehicles can differ either in their motion constraints or sensing capabilities. Approximation algorithms are useful for solving these routing problems because they produce solutions that can be efficiently computed and are relatively less sensitive to the noise in the data. In this paper, we present the first approximation algorithm for a 2-Depot, Heterogeneous Vehicle Routing Problem when the cost of direct travel between any pair of locations is no costlier than the cost of travel between the same locations and going through any intermediate location.

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

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: \( U_i (z) = C(z)\sum\nolimits_{j \in S_i } {(X_i - X_j - \tfrac{{L_{ij} z}} {{z - 1}})} \), 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.