Los actores y las iniciativas

Roadmap methods basically used to reduce the dimensionality of the environment in which path planning and navigation of a mobile robot takes place. Well known methods in this type are the Voronoi diagrams and the visibility graphs.

Voronoi Diagrams & Visibility Graphs

A voronoi diagram records information about the distances between sets of points in any dimensional space. For path planning, Voronoi tends to be used in two dimensional space, where sets of points all lie within a plane. A plane is divided into cells so that each cell contains exactly one site. For every point in the cell, the Euclidean distance of the point to the site within the cell must be smaller than the distance of that point to any other site in the

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plane. If this rule is followed across the entire plane, then the boundaries of the cells known as Voronoi edges will represent the point’s equidistance from the nearest 2 sites. The point where multiple boundaries meet, called a Voronoi vertex, is equidistance from its 3 nearest sites. To find the generalized Voronoi diagram, it is necessary to compute the diagram exactly or use an approximation based on the simpler problem of computing the Voronoi diagram for a set of discrete points. O’Dunlaing and Yap [82] have been proposed the use of generalized voronoi diagram for motion planning in autonomous robots. Later many improvised models have been proposed. Ilhan and Howie [83] have been addressed a new sensor roadmap model called generalized Voronoi graph incremental construction, which was included with an already existing incremental construction procedure.

Vlassis et. al. [84] have been proposed a method which generated a Voronoi graph by the robot dynamically builds of its configuration space by applying an adaptive kk-means clustering algorithm to the robot’s free space. They developed a Voronoi diagram of the robot’s free space by the probabilistic clustering plan. The probabilistic growing cell structures algorithm includes i) Initialization, ii) Adaptation, iii) updating, iv) Cluster insertion and v) Cluster deletion. Later triangulation method applied to the resulting cluster centers to construct a graph. Lisien and Morales [85] have been presented a new map designed for robots operating in large free space and possibly in higher dimensions called hierarchical atlas. This hierarchical atlas has two levels: at the highest level there is a topological map that creates the free space into submaps at the lower level. The lower-level submaps are simply a collection of features. For other tasks such as navigation, obstacle avoidance, and global localization the resulting map is also useful. Lee and Howie [86] have been introduced a new roadmap which is used to guide a convex body to explore an unknown planar workspace called the convex hierarchical generalized Voronoi graph (convex-HGVG). The convex-HGVG composed of two components: 1) convex-GVG edges, which are three- way equidistant; and 2) convex-R edges, which are two-way equidistant paths that connect disconnected convex-GVG edges.

Visibility graphs

The visibility graph of a polygon is a graph whose nodes relate to their vertices of the polygon, and whose edges correspond to the edges of the polygon formed by connecting the vertices that can see each other. Visibility graphs may be used to find the shortest Euclidean paths among a set of polygonal obstacles in the environment. The shortest path between two obstacles except at the vertices of the obstacles follows straight line segments. So the Euclidean shortest path is the shortest path in a visibility graph is the start and destination

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points and the vertices of the obstacles. Therefore, the Euclidean shortest path problem may be solved in the manner: constructing the visibility graph then apply a shortest path algorithm such as Dijkstra's algorithm to the graph. For planning the motion of a robot that has non- negligible size compared to the obstacles, a similar approach may be used after expanding the obstacles to compensate for this size of the robot.

Yamamoto et. al. [87] have proposed a near-time-optimal trajectory planning for car-like robots, where the connectivity graph is generated in a fashion very similar to that of speed optimization algorithm as proposed by Lozano-Perez & Wesley [88]. Oommen et. al. [89] have been presented an algorithm for navigation process composed of a number of travels and each traversal is from an arbitrary source point to an arbitrary destination point. In any stage of knowledge, a learned visibility graph is represented as the partially learned terrain model. After each traversal, this visibility graph is updated. It was proven that when the source and destination points are chosen randomly, the learned visibility graph converges to the visibility graph with probability one. Ultimately, optimized global path plan of the mobile robot was enabled by the availability of the complete visibility graph and it also eliminates the further usage of sensors. This Optimization technique includes the following theorems: Theorem1: The procedure NAVIGATE-LOCAL always finds a path from S to D in finite time for a non-interlocking workspace.

Theorem2: If one point exists in a non-interlocking workspace, the procedure BACKTRACK leads to a solution to the navigation problem.

Theorem 3: If the LVG converges to the VG with a probability one then no point in the free space has zero probability measure of being a source or destination point or a point on a path of traversal.

Theorem4: If N is the total number of vertices of the obstacles then the number of sensor operations performed within the procedure Update-Vgraph to learn the complete VG is N2. The visibility graph of a simple polygon has the polygon's vertices as its point locations, and the exterior of the polygon as the only obstacle. Visibility graphs of simple polygons must be Hamiltonian graphs: the boundary of the polygon forms a Hamiltonian cycle in the visibility graph. However, the precise characterization of these graphs is unknown.

It is a major open problem in this area whether there exists a polynomial time algorithm that can take input as a graph (possibly together with a fixed Hamiltonian in the cycle corresponds to the boundary) and produce output as a polygon for the visibility graph, if such a polygon exists.

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