Gaussian sampling [132]

The Gaussian sampling strategy follows some of the same motivation for sampling on the boundary. In this case, the goal is to obtain points near $\partial{\cal C}_{free}$ by using a Gaussian distribution that biases the samples to be closer to $\partial{\cal C}_{free}$, but the bias is gentler, as prescribed by the variance parameter of the Gaussian. The samples are generated as follows. Generate one sample, $q_1 \in {\cal C}$, uniformly at random. Following this, generate another sample, $q_2 \in {\cal C}$, according to a Gaussian with mean $q_1$; the distribution must be adapted for any topological identifications and/or boundaries of ${\cal C}$. If one of $q_1$ or $q_2$ lies in ${\cal C}_{free}$ and the other lies in ${\cal C}_{obs}$, then the one that lies in ${\cal C}_{free}$ is kept as a vertex in the roadmap. For some examples, this dramatically prunes the number of required vertices.