136 research outputs found
An Efficient Algorithm for Computing High-Quality Paths amid Polygonal Obstacles
We study a path-planning problem amid a set of obstacles in
, in which we wish to compute a short path between two points
while also maintaining a high clearance from ; the clearance of a
point is its distance from a nearest obstacle in . Specifically,
the problem asks for a path minimizing the reciprocal of the clearance
integrated over the length of the path. We present the first polynomial-time
approximation scheme for this problem. Let be the total number of obstacle
vertices and let . Our algorithm computes in time
a path of total cost
at most times the cost of the optimal path.Comment: A preliminary version of this work appear in the Proceedings of the
27th Annual ACM-SIAM Symposium on Discrete Algorithm
Efficient motion planning for problems lacking optimal substructure
We consider the motion-planning problem of planning a collision-free path of
a robot in the presence of risk zones. The robot is allowed to travel in these
zones but is penalized in a super-linear fashion for consecutive accumulative
time spent there. We suggest a natural cost function that balances path length
and risk-exposure time. Specifically, we consider the discrete setting where we
are given a graph, or a roadmap, and we wish to compute the minimal-cost path
under this cost function. Interestingly, paths defined using our cost function
do not have an optimal substructure. Namely, subpaths of an optimal path are
not necessarily optimal. Thus, the Bellman condition is not satisfied and
standard graph-search algorithms such as Dijkstra cannot be used. We present a
path-finding algorithm, which can be seen as a natural generalization of
Dijkstra's algorithm. Our algorithm runs in time, where~ and are the number of vertices and
edges of the graph, respectively, and is the number of intersections
between edges and the boundary of the risk zone. We present simulations on
robotic platforms demonstrating both the natural paths produced by our cost
function and the computational efficiency of our algorithm
Geometric Secluded Paths and Planar Satisfiability
We consider paths with low exposure to a 2D polygonal domain, i.e., paths which are seen as little as possible; we differentiate between integral exposure (when we care about how long the path sees every point of the domain) and 0/1 exposure (just counting whether a point is seen by the path or not). For the integral exposure, we give a PTAS for finding the minimum-exposure path between two given points in the domain; for the 0/1 version, we prove that in a simple polygon the shortest path has the minimum exposure, while in domains with holes the problem becomes NP-hard. We also highlight connections of the problem to minimum satisfiability and settle hardness of variants of planar min- and max-SAT
Near-Optimal Min-Sum Motion Planning for Two Square Robots in a Polygonal Environment
Let be a planar polygonal environment
(i.e., a polygon potentially with holes) with a total of vertices, and let
be two robots, each modeled as an axis-aligned unit square, that can
translate inside . Given source and target placements
of and , respectively, the goal is to
compute a \emph{collision-free motion plan} , i.e., a motion
plan that continuously moves from to and from to
so that and remain inside and do not collide with
each other during the motion. Furthermore, if such a plan exists, then we wish
to return a plan that minimizes the sum of the lengths of the paths traversed
by the robots, . Given and a parameter , we present an
-time -approximation algorithm
for this problem. We are not aware of any polynomial time algorithm for this
problem, nor do we know whether the problem is NP-Hard. Our result is the first
polynomial-time -approximation algorithm for an optimal motion
planning problem involving two robots moving in a polygonal environment.Comment: The conference version of the paper is accepted to SODA 202
Exact Computation of a Manifold Metric, via Lipschitz Embeddings and Shortest Paths on a Graph
Data-sensitive metrics adapt distances locally based the density of data
points with the goal of aligning distances and some notion of similarity. In
this paper, we give the first exact algorithm for computing a data-sensitive
metric called the nearest neighbor metric. In fact, we prove the surprising
result that a previously published -approximation is an exact algorithm.
The nearest neighbor metric can be viewed as a special case of a
density-based distance used in machine learning, or it can be seen as an
example of a manifold metric. Previous computational research on such metrics
despaired of computing exact distances on account of the apparent difficulty of
minimizing over all continuous paths between a pair of points. We leverage the
exact computation of the nearest neighbor metric to compute sparse spanners and
persistent homology. We also explore the behavior of the metric built from
point sets drawn from an underlying distribution and consider the more general
case of inputs that are finite collections of path-connected compact sets.
The main results connect several classical theories such as the conformal
change of Riemannian metrics, the theory of positive definite functions of
Schoenberg, and screw function theory of Schoenberg and Von Neumann. We develop
novel proof techniques based on the combination of screw functions and
Lipschitz extensions that may be of independent interest.Comment: 15 page
Coverage and Time-optimal Motion Planning for Autonomous Vehicles
Autonomous vehicles are rapidly advancing with a variety of applications, such as area surveillance, environment mapping, and intelligent transportation. These applications require coverage and/or time-optimal motion planning, where the major challenges include uncertainties in the environment, motion constraints of vehicles, limited energy resources and potential failures. While dealing with these challenges in various capacities, this dissertation addresses three fundamental motion planning problems: (1) single-robot complete coverage in unknown environment, (2) multi-robot resilient and efficient coverage in unknown environment, and (3) time-optimal risk-aware motion planning for curvature-constrained vehicles.
First, the ε* algorithm is developed for online coverage path planning in unknown environment using a single autonomous vehicle. It is computationally efficient, and can generate the desired back-and-forth path with less turns and overlappings. ε* prevents the local extrema problem, thus can guarantee complete coverage. Second, the CARE algorithm is developed which extends ε* for multi-robot resilient and efficient coverage in unknown environment. In case of failures, CARE guarantees complete coverage via dynamic task reallocations of other vehicles, hence provides resilience. Moreover, it reallocates idling vehicles to support others in their tasks, hence improves efficiency. Finally, the T* algorithm is developed to find the time-optimal risk-aware path for curvature-constrained vehicles. We present a novel risk function based on the concept of collision time, and integrate it with the time cost for optimization.
The above-mentioned algorithms have been validated via simulations in complex scenarios and/or real experiments, and the results have shown clear advantages over existing popular approaches
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