Crc Press - Mechanical Engineering Handbook - Robotics 1 Part 8 pptx

14-70 Section 14 © 1999 by CRC Press LLC The abstract model of the path-planning problem can take different forms depending on the application characteristics. It can be in two dimensions or in three dimensions. The concern can be the end effector alone or the entire robot arm. The end effector can be considered as a solid body or an infinitesimal point. Different considerations can have significant implications on the solution methodology and complexity. In this handbook, we will only discuss the simplest cases in which a point end effector in two-dimensional space is concerned. The readers are referred to Latombe (1991) for more complex procedures. Road Map Approach Based on Visibility Graph The road map approach is one of the earliest path-planning methods. The obstacles are modeled as polygons. A visibility graph is a nondirected graph. The nodes of the graph are the vertices of the polygons, the initial point and the goal point. The links of the graphs are straight-line segments that connect a pair of nodes without intersecting with any obstacles. A reduced visibility graph for the example is shown in Figure 14.8.2. A reduced visibility graph does not contain links that are dominated by other links in terms of distance. The elliptical obstacle is approximated by a hexagon. In the visibility graph, all the paths consisting of successive links that connect C init to C goal represent semicollision-free paths. The coarse line represents one of these paths. The use of the term “semicollision free” is due to the fact the path may actually contact an obstacle. It is clear that the path is not unique. In the example the possible paths can be C init AC goal , C init BC goal , or C init CD goal . Some offer shorter travel distances while others offer smoother paths. This method can be extended to include circular end effector and obstacles which have lines and arcs as boundaries. Road Map Approach Based on Voronoi Diagram For the same problem described above, one can create a Voronoi diagram based on the vertices and line segments of the obstacles and the working envelope and use this graph to generate a collision-free path. A Voronoi diagram is a diagram that consists of lines having equal distance from the adjacent objects. Obviously, the Voronoi diagram does not touch the obstacles and can provide collision-free paths. A Voronoi diagram in a polygonal space with polygonal obstacles is composed of straight line segments and parabolas. When both adjacent object segments are straight lines or vertices, the segment of the Voronoi diagram is a straight line. When one object segment is a point while the other is a line segment, the segment of Voronoi diagram is a parabola. Two additional links need to be created to connect the C init and C goal to the Voronoi diagram. Any set of links that connects C init and C goal through the diagram represents a collision-free path. Unlike the road map approach based on visibility graph, this approach tends to maximize the clearance between the robot and the obstacles. For the characteristics and creation of the Voronoi diagrams, the reader is referred to Okabe et al. (1992). FIGURE 14.8.1Illustration of path-planning problem. . 1 4-7 0 Section 14 © 19 99 by CRC Press LLC The abstract model of the path-planning problem can take different forms depending on the. and creation of the Voronoi diagrams, the reader is referred to Okabe et al. (19 92). FIGURE 14 .8. 1Illustration of path-planning problem. . and complexity. In this handbook, we will only discuss the simplest cases in which a point end effector in two-dimensional space is concerned. The readers are referred to Latombe (19 91) for more complex procedures. Road