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### Motion Planningin Virtual Environments

Dan HalperinYesha Sivan

TA: Alon Shalita

Spring 2007

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

[Lozano-Peréz ’79]

- 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

Point robot

www.seas.upenn.edu/~jwk/motionPlanning

c7

c4

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c15

c11

c1

c10

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c1

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c5

c13

c3

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Connectivity graphwww.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

What’s behind the maze solver that we saw last week:

translational motion planning for a polygon

among polygos using exact Minkowski sums

=

Planar Minkowski sumsGiven two sets A and B in the plane, their Minkowski sum, denoted AB, is:

A B = {a + b | a A, b B}

Convex-convex

We are given two polygons P and Q with mandn 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.

When at least one is non-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).

P1

Q

P2

P

Q2

PQ

The decomposition methodThe 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 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)

Key issues

- Collision checking
- Node sampling
- Finding nearby nodes
- Node connection

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