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Motion Planning i n Virtual Environments. Dan Halperin Yesha Sivan TA: Alon Shalita. Spring 2007. Basics of Motion Planning (D.H.). Motion planning: the basic problem.

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Motion planning i n virtual environments

Motion Planningin Virtual Environments

Dan HalperinYesha Sivan

TA: Alon Shalita

Spring 2007

Basics of Motion Planning (D.H.)


Motion planning the basic problem
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 space of a robot system with k degrees of freedom
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
Point robot

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


Trapezoidal decomposition

c14

c7

c4

c5

c8

c2

c15

c11

c1

c10

c13

c3

c9

c6

c12

Trapezoidal decomposition

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


Connectivity graph

c14

c7

c4

c5

c8

c2

c15

c11

c1

c10

c11

c8

c2

c3

c4

c6

c1

c7

c9

c5

c13

c3

c9

c6

c12

c14

c12

c10

c13

c15

Connectivity graph

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


Two major planning frameworks
Two major planning frameworks

  • Cell decomposition

  • Road map

  • Motion planning methods differ along additional parameters


Hardness
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
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


Planar minkowski sums

=

Planar 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}


Convex convex
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
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).


The decomposition method

Q1

P1

Q

P2

P

Q2

PQ

The decomposition method

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.



What is the number of dof s
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 roadmaps

milestone complicated robots?

qg

qb

Probabilistic roadmaps

free space

[Kavraki, Svetska, Latombe,Overmars, 95]


Key issues
Key issues complicated robots?

  • Collision checking

  • Node sampling

  • Finding nearby nodes

  • Node connection


THE END complicated robots?


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