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Spring 2007
Motion Planning in Virtual Environments
Dan Halperin Yesha Sivan
TA: Alon Shalita
Basics of Motion Planning (D.H.)
Motion planning:the basic problem
Let B be a system (the robot) with k degrees of freedom moving in a known environment cluttered with obstacles. Given free start and goal placements for B decide whether there is a collision free motion for B from start to goal and if so plan such a motion.
Configuration spaceof a robot system with k degrees of freedom
the space of parametric representation of all possible robot configurations
C-obstacles: the expanded obstacles the robot -> a point k dimensional space point in configuration space: free,
forbidden, semi-free path -> curve
[Lozano-Peréz ’79]
Point robot
www.seas.upenn.edu/~jwk/motionPlanning
Trapezoidal decomposition
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www.seas.upenn.edu/~jwk/motionPlanning
Connectivity graph
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www.seas.upenn.edu/~jwk/motionPlanning
Two major planning frameworks
Cell decomposition Road map
Motion planning methods differ along additional parameters
Hardness The problem is hard when k is part of the
input [Reif 79], [Hopcroft et al. 84], … [Reif 79]: planning a free path for a robot
made of an arbitrary number of polyhedral bodies connected together at some joint vertices, among a finite set of polyhedral obstacles, between any two given configurations, is a PSPACE-hard problem
Translating rectangles, planar linkages
A complete solution
roadmap [Canny 87]:a singly exponential solution, nk(log n)dO(k^2) expected time
What’s behind the maze solver that we saw last week:
translational motion planning for a polygon among polygos using exact Minkowski sums
Given two sets A and B in the plane, their Minkowski sum, denoted A B, is:
A B = {a + b | a A, b B}
Planar Minkowski sums
=
We are given two polygons P and Q with m and n vertices respectively. If both polygons are convex, the complexity of their sum is m + n, and we can compute it in (m + n) time using a very simple procedure.
Convex-convex
If only one of the polygons is convex, the complexity of their sum is (mn).
If both polygons are non-convex, the complexity of their sum is (m2n2).
When at least one is non-convex
The prevailing method for computing the sum of two non-convex polygons: Decompose P and Q into convex sub- polygons, compute the pair-wise sums of the sub-polygons and obtain the union of these sums.
The decomposition method
P
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Q2P Q
The maze solver that we saw last week uses CGAL’s Minkowski sum package
What is the number of DoF’s?
a polygon robot translating in the plane
a polygon robot translating and rotating
a spherical robot moving in space a spatial robot translating and
rotating a snake robot in the plane with 3 links
How to cope with many degrees of freedom and more complicated robots?
prevalent methods: sampling-based planners
We start with the archetype: probabilistic roadmap (PRM)
Probabilistic roadmapsProbabilistic roadmapsfree space
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milestone
[[Kavraki, Svetska, Latombe,OvermarsKavraki, Svetska, Latombe,Overmars, 95], 95]
Key issues
Collision checking Node sampling Finding nearby nodes Node connection
THE END
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