The sampling-based planning algorithms in Chapter 5 were designed to terminate upon finding a solution path. In the current setting, termination is complicated by the fact that we are interested in solutions from all initial configurations. Since $ {\alpha }$ is dense, the volume of uncovered points in $ {\cal C}_{free}$ tends to zero. After some finite number of iterations, it would be nice to measure the quality of the approximation and then terminate when the desired quality is achieved. This was also possible with the visibility sampling-based roadmap in Section 5.6.2. Using random samples, an estimate of the fraction of $ {\cal C}_{free}$ can be obtained by recording the percentage of failures in obtaining a sample in $ {\cal C}_{free}$ that is outside of the cover. For example, if a new neighborhood is created only once in $ 1000$ iterations, then it can be estimated that $ 99.9$ percent of $ {\cal C}_{free}$ is covered. High-probability bounds can also be determined. Termination conditions are given in [983] that ensure with probability greater than $ P_c$ that at least a fraction $ \alpha \in (0,1)$ of $ {\cal C}_{free}$ has been covered. The constants $ P_c$ and $ \alpha$ are given as parameters to the algorithm, and it will terminate when the condition has been satisfied using rigorous statistical tests. If deterministic sampling is used, then termination can be made to occur based on the dispersion, which indicates the largest ball in $ {\cal C}_{free}$ that does not contain the center of another neighborhood. One problem with volume-based criteria, such as those suggested here, is that there is no way to ensure that the cover preserves the connectivity of $ {\cal C}_{free}$. If two portions of $ {\cal C}_{free}$ are connected by a narrow passage, the cover may miss a neighborhood that has very small volume yet is needed to connect the two portions.

Figure 8.18: (a) A approximate cover for a 2D configuration space. (b) Level sets of a navigation function
...,height=2.6in,width=2.6in} \\
(a) & (b)

Example 8..18 (2D Example of Computed Funnels)   Figure 8.18 shows a 2D example that was computed using random samples and the algorithm in Figure 8.17. Note that once a cover is computed, it can be used to rapidly compute different navigation functions and vector fields for various goals. This example is mainly for illustrative purposes. For the case of a polygonal environment, constructing covers based on convex polygons would be more efficient. $ \blacksquare$

Steven M LaValle 2012-04-20