Orourkech8

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: UMass Lowell Computer Science 91.504 Advanced Algorithms Computational Geometry Prof. Karen Daniels Spring, 2010 O'Rourke Chapter 8 Motion Planning Chapter 8 Motion Planning Shortest Paths (robot = point) Moving a Disk Translating a Convex Polygon Moving a Ladder Robot Arm Motion Separability Problem Specification s t Assume fixed environment of impenetrable objects e.g. Polygons or polyhedra Robot: movable object at initial position e.g. point robot starting at s Goal: Plan motion to final position (e.g. t) avoiding robot penetration of objects Sliding contact is acceptable Questions: Does there exist a "free" (collision-avoiding) path? How to construct such a path? How to find shortest such path? Shortest Paths s t Reduce possibilities to finite list! Shortest path segment endpoint vertex of obstacle (consider s,t as "point obstacles) Shortest path is subpath of visibility graph of vertices of obstacle polygons visibility graph requires (n2) time Algorithm: DIJKSTRA'S ALGORITHM T {s} while t not in T do Find edge e in (G \ T) that augments T to reach a node x whose distance from s is minimum T T + {e} Assume: - Polygonal obstacles have total of n vertices - Points s, t are outside obstacles - Points s, t are degenerate polygons - Obstacles are disjoint Moving a Disk Revised Goal: find any path to t not necessarily shortest path Shrink disk to a point Grow obstacles by disk radius Form union of grown obstacles If t, s in same component of plane, there is a free path t s t find it by modifying visibility graph to include circular arcs from grown obstacles s O(n2 lg n) Why Does it Work? Minkowski Sum: vector sum for point sets A B = {a + b | a A, b B} a+b a b a1 a4 a 3 a2 a4+b3 a3+b2 b2 A B Simple Case b3 A B a2+b1 b1 Convex/Convex Case Why Does it Work? (continued) (a5 + b4 ), (a5 + b3 ) (a3 + b4 ), (a4 + b4 ) a4 a5 a3 a1 a2 a5+b4 a4+b4 a3+b4 a5+b3 a2+b3 b4 b3 b2 b1 a3+b2 a3+b1 a2+b2 A B (a3 + b1 ), (a3 + b2 ) (a2 + b2 ), (a2 + b3 ) Polygonal Nonconvex/Nonconvex Case Minkowski Sum: Some Properties Shape is translationally invariant ( A + t ) B = ( A B) + t = A ( B + t ) Commutative A B = B A Union formulation A B = aA,bB {a + b} = {a + B} = {b + A} aA bB When A, B convex, sum is convex Minkowski Sum: Properties (continued) TRANSLATIONAL INTERSECTION ( B + t ) A iff t ( A (- B )) Why?? t a b Consider Simple Case: a=A, b=B b+t = a t = a -b Note: -b = b rotated by Mi...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern