The Cost of Bounded Curvature

We study the motion-planning problem for a car-like robot whose turning radius is bounded from below by one and which is allowed to move in the forward direction only (Dubins car). For two robot configurations $\sigma, \sigma'$, let $\ell(\sigma, \sigma')$ be the shortest bounded-curvature path from $\sigma$ to $\sigma'$. For $d \geq 0$, let $\ell(d)$ be the supremum of $\ell(\sigma, \sigma')$, over all pairs $(\sigma, \sigma')$ that are at Euclidean distance $d$. We study the function \dub(d) = \ell(d) - d, which expresses the difference between the bounded-curvature path length and the Euclidean distance of its endpoints. We show that \dub(d) decreases monotonically from \dub(0) = 7\pi/3 to \dub(\ds) = 2\pi, and is constant for d \geq \ds. Here \ds \approx 1.5874. We describe pairs of configurations that exhibit the worst-case of \dub(d) for every distance $d$

Approximation of Curvature-constrained Shortest Paths through a Sequence of Points

Let $B$ be a point robot moving in the plane, whose path is constrained to forward motions with curvature at most 1, and let \X denote a sequence of $n$ points. Let $s$ be the length of the shortest curvature-constrained path for $B$ that visits the points of \X in the given order. We show that if the points of \X are given \emph{on-line} and the robot has to respond to each point immediately, there is no strategy that guarantees a path whose length is at most~$f(n)s$, for any finite function~$f(n)$. On the other hand, if all points are given at once, a path with length at most $5.03 s$ can be computed in linear time. In the \emph{semi-online} case, where the robot not only knows the next input point but is able to ``see'' the future input points included in the disk with radius $R$ around the robot, a path of length $(5.03 + O(1/R))s$ can be computed

Bounded-Curvature Shortest Paths through a Sequence of Points

We consider the problem of computing shortest paths having curvature at most one almost everywhere and visiting a sequence of $n$ points in the plane in a given order. This problem arises naturally in path planning for point car-like robots in the presence of polygonal obstacles, and is also a sub-problem of the Dubins Traveling Salesman Problem. This problem reduces to minimizing the function $F:\R^n\rightarrow\R$ that maps $(\theta_1,\ldots,\theta_n)$ to the length of a shortest curvature-constrained path that visits the points $p_1, \ldots, p_n$ in order and whose tangent in $p_i$ makes an angle $\theta_i$ with the $x$-axis. We show that when consecutive points are distance at least $4$ apart, all minima of $F$ are realized over at most $2^k$ disjoint convex polyhedra over which $F$ is strictly convex; each polyhedron is defined by $4n-1$ linear inequalities and $k$ denotes, informally, the number of $p_i$ such that the angle $\angle(p_{i-1},p_i,p_{i+1})$ is small. A curvature-constrained shortest path visiting a sequence points can therefore be approximated by standard convex optimization methods, which presents an interesting alternative to the known polynomial-time algorithms that can only compute a multiplicative constant factor approximation. Our technique also opens new perspectives for bounded-curvature path planning among polygonal obstacles. In particular, we show that, under certain conditions, if the sequence of points where a shortest path touches the obstacles is known then ``connecting the dots'' reduces to a family of convex optimization problems

Performance optimization for unmanned vehicle systems

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2008.Includes bibliographical references (p. 149-157).Technological advances in the area of unmanned vehicles are opening new possibilities for creating teams of vehicles performing complex missions with some degree of autonomy. Perhaps the most spectacular example of these advances concerns the increasing deployment of unmanned aerial vehicles (UAVs) in military operations. Unmanned Vehicle Systems (UVS) are mainly used in Information, Surveillance and Reconnaissance missions (ISR). In this context, the vehicles typically move about a low-threat environment which is sufficiently simple to be modeled successfully. This thesis develops tools for optimizing the performance of UVS performing ISR missions, assuming such a model.First, in a static environment, the UVS operator typically requires that a vehicle visit a set of waypoints once or repetitively, with no a priori specified order. Minimizing the length of the tour traveled by the vehicle through these waypoints requires solving a Traveling Salesman Problem (TSP). We study the TSP for the Dubins' vehicle, which models the limited turning radius of fixed wing UAVs. In contrast to previously proposed approaches, our algorithms determine an ordering of the waypoints that depends on the model of the vehicle dynamics. We evaluate the performance gains obtained by incorporating such a model in the mission planner.With a dynamic model of the environment the decision making level of the UVS also needs to solve a sensor scheduling problem. We consider M UAVs monitoring N > M sites with independent Markovian dynamics, and treat two important examples arising in this and other contexts, such as wireless channel or radar waveform selection. In the first example, the sensors must detect events arising at sites modeled as two-state Markov chains. In the second example, the sites are assumed to be Gaussian linear time invariant (LTI) systems and the sensors must keep the best possible estimate of the state of each site.(cont.) We first present a bound on the achievable performance which can be computed efficiently by a convex program, involving linear matrix inequalities in the LTI case. We give closed-form formulas for a feedback index policy proposed by Whittle. Comparing the performance of this policy to the bound, it is seen to perform very well in simulations. For the LTI example, we propose new open-loop periodic switching policies whose performance matches the bound.Ultimately, we need to solve the task scheduling and motion planning problems simultaneously. We first extend the approach developed for the sensor scheduling problems to the case where switching penalties model the path planning component. Finally, we propose a new modeling approach, based on fluid models for stochastic networks, to obtain insight into more complex spatiotemporal resource allocation problems. In particular, we give a necessary and sufficient stabilizability condition for the fluid approximation of the problem of harvesting data from a set of spatially distributed queues with spatially varying transmission rates using a mobile server.by Jerome Le Ny.Ph.D