Unformatted text preview: nkowski Sum: Properties
(continued) TRANSLATIONAL INTERSECTION ( B + t ) A iff t ( A (- B )) A (- B)
-B t B A B B Polygon Motion Planning To plan motion for a shape P amidst polygonal obstacles U set of all displacements of P relative to U U ( - P ) such that (translated) P intersects U: set of all displacements of P relative to U such that (translated) P does not intersect U ( - P ) U: If t, s in same component of plane, there is a free path
t find it by modifying visibility graph to include circular arcs from grown obstacles U t U P s s Minkowski Sum Algorithms
Algorithms for Constructing the Minkowski Sum of 2 Polygons A and B A [n vertices] Convex Convex NonConvex B [m vertices] Convex NonConvex NonConvex Size O(n+m) O(nm) O(n2m2) Running Time O(n+m) O(nm) O(n2m2) Convex A, B NonConvex A, B
a5 a4 a a supports b b 3 3 4 3 a1 a2 b4 b3 b2 b3 supports a2 a3 merge edge copies in slope order b1 Identify vertex/edge support pairs Minkowski Sum Exercises
These statements are about Minkowski sums for 2 2D point sets A and B:
(a) provide a counterexample that shows this is false: A B = A (- B)
B -B (b) prove this is true B - ((- A) B ) = A (- B )
B Moving a Ladder Rotation adds degree of freedom - makes the "configuration space" 3D Two Methods to find free path through configuration space: 1) Cell decomposition 2) Retraction
Source: O'Rourke Configuration space if robot and obstacles can have circular arcs (smoothly joined) Source: PhD dissertation by Steven Cy Trac, U. Miami Moving a Ladder through Polygonal Obstacles (continued) Find Free Path through Configuration Space Cell decomposition method Shaded regions comprise "free space."
No path in G0 from A to C. disconnect Partition configuration space into finite number of "well-behaved cells" For a single orientation Cell = connected region in free space of appropriate configuration space Connectivity graph "dual graph" G represents cell structure Identify O(n2) critical orientations where combinatorial structure of connectivity graph changes Alignment of ladder with either obstacle edges or 2 obstacle vertices. Form overall connectivity graph reference point obstacles Disconnect for this angle too! Determine...
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