Supplementary On-line Proofs for Sampling-Diagram Automata: a Tool for Analyzing Path Quality in Tree Planners

Note that the main lemma used in the proof of Theorem 2 is Lemma 7 below, and the other lemmas are used as auxiliary geometric lemmas. For completeness , we include in this document a figure of the Promenade motion-planning problem, as it appears in the main text. q1 q2 A1 A2 B1 B2 α α 1 Fig. 1: An illustration of the Promenade motion planning problem, a square obstacle within a square bounding-box. In this example, q1 and q2 lie on opposite sides of the promenade. Type-A (solid line) and type-B (dashed line) solution paths between q1 and q2 are shown, as defined in Section 4. The ratio between their length, denoted µ, is approximately 1 3 in this case.