173 research outputs found
Meeting in a Polygon by Anonymous Oblivious Robots
The Meeting problem for searchers in a polygon (possibly with
holes) consists in making the searchers move within , according to a
distributed algorithm, in such a way that at least two of them eventually come
to see each other, regardless of their initial positions. The polygon is
initially unknown to the searchers, and its edges obstruct both movement and
vision. Depending on the shape of , we minimize the number of searchers
for which the Meeting problem is solvable. Specifically, if has a
rotational symmetry of order (where corresponds to no
rotational symmetry), we prove that searchers are sufficient, and
the bound is tight. Furthermore, we give an improved algorithm that optimally
solves the Meeting problem with searchers in all polygons whose
barycenter is not in a hole (which includes the polygons with no holes). Our
algorithms can be implemented in a variety of standard models of mobile robots
operating in Look-Compute-Move cycles. For instance, if the searchers have
memory but are anonymous, asynchronous, and have no agreement on a coordinate
system or a notion of clockwise direction, then our algorithms work even if the
initial memory contents of the searchers are arbitrary and possibly misleading.
Moreover, oblivious searchers can execute our algorithms as well, encoding
information by carefully positioning themselves within the polygon. This code
is computable with basic arithmetic operations, and each searcher can
geometrically construct its own destination point at each cycle using only a
compass. We stress that such memoryless searchers may be located anywhere in
the polygon when the execution begins, and hence the information they initially
encode is arbitrary. Our algorithms use a self-stabilizing map construction
subroutine which is of independent interest.Comment: 37 pages, 9 figure
Meeting in a Polygon by Anonymous Oblivious Robots
The Meeting problem for k>=2 searchers in a polygon P (possibly with holes) consists in making the searchers move within P, according to a distributed algorithm, in such a way that at least two of them eventually come to see each other, regardless of their initial positions. The polygon is initially unknown to the searchers, and its edges obstruct both movement and vision. Depending on the shape of P, we minimize the number of searchers k for which the Meeting problem is solvable. Specifically, if P has a rotational symmetry of order sigma (where sigma=1 corresponds to no rotational symmetry), we prove that k=sigma+1 searchers are sufficient, and the bound is tight. Furthermore, we give an improved algorithm that optimally solves the Meeting problem with k=2 searchers in all polygons whose barycenter is not in a hole (which includes the polygons with no holes). Our algorithms can be implemented in a variety of standard models of mobile robots operating in Look-Compute-Move cycles. For instance, if the searchers have memory but are anonymous, asynchronous, and have no agreement on a coordinate system or a notion of clockwise direction, then our algorithms work even if the initial memory contents of the searchers are arbitrary and possibly misleading. Moreover, oblivious searchers can execute our algorithms as well, encoding information by carefully positioning themselves within the polygon. This code is computable with basic arithmetic operations (provided that the coordinates of the polygon\u27s vertices are algebraic real numbers in some global coordinate system), and each searcher can geometrically construct its own destination point at each cycle using only a compass. We stress that such memoryless searchers may be located anywhere in the polygon when the execution begins, and hence the information they initially encode is arbitrary. Our algorithms use a self-stabilizing map construction subroutine which is of independent interest
Algorithms for Optimizing Search Schedules in a Polygon
In the area of motion planning, considerable work has been done on guarding
problems, where "guards", modelled as points, must guard a polygonal
space from "intruders". Different variants
of this problem involve varying a number of factors. The guards performing
the search may vary in terms of their number, their mobility, and their
range of vision. The model of intruders may or may not allow them to
move. The polygon being searched may have a specified starting point,
a specified ending point, or neither of these. The typical question asked
about one of these problems is whether or not certain polygons can be
searched under a particular guarding paradigm defined by the types
of guards and intruders.
In this thesis, we focus on two cases of a chain of guards searching
a room (polygon with a specific starting point) for mobile intruders.
The intruders must never be allowed to escape through the door undetected.
In the case of the two guard problem, the guards must start at the door
point and move in opposite directions along the boundary of the
polygon, never crossing the door point. At all times, the
guards must be able to see each other. The search is complete once both
guards occupy the same spot elsewhere on the polygon. In the case of
a chain of three guards, consecutive guards in the chain must always
be visible. Again, the search starts at the door point, and the outer
guards of the chain must move from the door in opposite directions.
These outer guards must always remain on the boundary of the polygon.
The search is complete once the chain lies entirely on a portion of
the polygon boundary not containing the door point.
Determining whether a polygon can be searched is a problem in the area
of visibility in polygons; further to that, our work is related
to the area of planning algorithms. We look for ways to find optimal schedules that minimize
the distance or time required to complete the search. This is done
by finding shortest paths in visibility diagrams that indicate valid
positions for the guards. In the case of
the two-guard room search, we are able to find the shortest distance
schedule and the quickest schedule. The shortest distance schedule
is found in O(n^2) time by solving an L_1 shortest path problem
among curved obstacles in two dimensions. The quickest search schedule is
found in O(n^4) time by solving an L_infinity shortest path
problem among curved obstacles in two dimensions.
For the chain of three guards, a search schedule minimizing the total
distance travelled by the outer guards is found in O(n^6) time by
solving an L_1 shortest path problem among curved obstacles in two dimensions
Coverage & cooperation: Completing complex tasks as quickly as possible using teams of robots
As the robotics industry grows and robots enter our homes and public spaces, they are increasingly expected to work in cooperation with each other. My thesis focuses on multirobot planning, specifically in the context of coverage robots, such as robotic lawnmowers and vacuum cleaners.
Two problems unique to multirobot teams are task allocation and search. I present a task allocation algorithm which balances the workload amongst all robots in the team with the objective of minimizing the overall mission time. I also present a search algorithm which robots can use to find lost teammates. It uses a probabilistic belief of a target robot’s position to create a planning tree and then searches by following the best path in the tree.
For robust multirobot coverage, I use both the task allocation and search algorithms. First the coverage region is divided into a set of small coverage tasks which minimize the number of turns the robots will need to take. These tasks are then allocated to individual robots. During the mission, robots replan with nearby robots to rebalance the workload and, once a robot has finished its tasks, it searches for teammates to help them finish their tasks faster
Optimal Online Escape Path Against a Certificate
More than fifty years ago Bellman asked for the best escape path within a known forest but for an unknown starting position. This deterministic finite path is the shortest path that leads out of a given environment from any starting point. There are some worst case positions where the full path length is required. Up to now such a fixed ultimate optimal escape path for a known shape for any starting position is only known for some special convex shapes (i.e., circles, strips of a given width, fat convex bodies, some isosceles triangles).
Therefore, we introduce a different, simple and intuitive escape path, the so-called certificate path which make use of some additional information w.r.t. the starting point s. This escape path depends on the starting position s and takes the distances from s to the outer boundary of the environment into account. Because of this, in the above convex examples the certificate path always (for any position s) leaves the environment earlier than the ultimate escape path.
Next we assume that neither the precise shape of the environment nor the location of the starting point is not known, we have much less information. For a class of environments (convex shapes and shapes with kernel positions) we design an online strategy that always leaves the environment. We show that the path length for leaving the environment is always shorter than 3.318764 the length of the corresponding certificate path. We also give a lower bound of 3.313126 which shows that for the above class of environments the factor 3.318764 is (almost) tight
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