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

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QP1

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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