A polyhedral C-space obstacle

Most of the previous ideas generalize nicely for the case of a polyhedral robot that is capable of translation only in a 3D world that contains polyhedral obstacles. If $ {\cal A}$ and $ {\cal O}$ are convex polyhedra, the resulting $ {\cal C}_{obs}$ is a convex polyhedron.

Figure 4.20: Three different types of contact, each of which generates a different kind of $ {\cal C}_{obs}$ face.
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Type FV & Type VF & Type EE \\

There are three different kinds of contacts that each lead to half-spaces in $ {\cal C}$:

  1. Type FV: A face of $ {\cal A}$ and a vertex of $ {\cal O}$
  2. Type VF: A vertex of $ {\cal A}$ and a face of $ {\cal O}$
  3. Type EE: An edge of $ {\cal A}$ and an edge of $ {\cal O}$ .
These are shown in Figure 4.20. Each half-space defines a face of the polyhedron, $ {\cal C}_{obs}$. The representation of $ {\cal C}_{obs}$ can be constructed in $ O(n + m + k)$ time, in which $ n$ is the number of faces of $ {\cal A}$, $ m$ is the number of faces of $ {\cal O}$, and $ k$ is the number of faces of $ {\cal C}_{obs}$, which is at most $ nm$ [411].

Steven M LaValle 2012-04-20