CS 326A: Motion Planning

Criticality-Based Motion Planning: Target Finding

Trapezoidal decomposition

Criticality-Based Planning

Define a property P Decompose the configuration space into

“regular” regions (cells) over which P is constant.

Use this decomposition for planning Issues:

- What is P? It depends on the problem- How to use the decomposition?

Approach is practical only in low-dimensional spaces:- Complexity of the arrangement of cells- Sensitivity to floating point errors

Topics of this class and the next one

Target finding Information (or belief) state/space

Assembly planning Path space

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Example

robot’s visibilityregion

hidingregion 1

cleared region

2 3

4 5 6

robot

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Problem

A target is hiding in an environment cluttered with obstacles

A robot or multiple robots with vision sensor must find the target

Compute a motion strategy with minimal number of robot(s)

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Assumptions

Target is unpredictable and can move arbitrarily fast

Environment is polygonal Both the target and robots are

modeled as points A robot finds the target when the

straight line joining them intersects no obstacles (omni-directional vision)

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Animated Target-Finding Strategy

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Does a solution always exist

for a single robot?

Hard to test:No “hole” in the workspace

Easy to test: “Hole” in the workspace

No !

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Effect of Geometry on the Number of Robots

Two robotsare needed

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Effect of Number n of EdgesMinimal number of robots N = (log n)

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Effect of Number h of Holes

N=Θ( h)

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Information State

Example of an information state = (x,y,a=1,b=1,c=0) An initial state is of the form (x,y,a=1,b=1,...,u=1) A goal state is any state of the form (x,y,a=0,b=0,...,

u=0)

(x,y)

a = 0 or 1

c = 0 or 1

b = 0 or 1

0 cleared region1 contaminated region

visibility region

obstacle edgefree edge

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Critical Line

a=0b=1

(x,y,a=0,b=1)

Information state is unchanged(x,y,a=0,b=1)

a=0b=1

a=0b=0

(x,y,a=0,b=0)Critical line

contaminated area

cleared area

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Grid-Based Discretization

Ignores critical lines Visits many “equivalent” states Many information states per grid point Potentially very inefficient

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Discretization into Conservative Cells

In each conservative cell, the “topology” of the visibilityregion remains constant, i.e., the robot keeps seeing the same obstacle edges

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Discretization into Conservative Cells

In each conservative cell, the “topology” of the visibilityregion remains constant, i.e., the robot keeps seeing the same obstacle edges

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Discretization into Conservative Cells

In each conservative cell, the “topology” of the visibilityregion remains constant, i.e., the robot keeps seeing the same obstacle edges

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Search Graph {Nodes} = {Conservative Cells} X {Information

States}

Node (c,i) is connected to (c’,i’) iff:• Cells c and c’ share an edge (i.e., are adjacent)• Moving from c, with state i, into c’ yields state i’

Initial node (c,i) is such that:• c is the cell where the robot is initially located• i = (1, 1, …, 1)

Goal node is any node where the information state is (0, 0, …, 0)

Size is exponential in the number of edges

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Example

A

B C D

E

(C,a=1,b=1)

(D,a=1)(B,b=1)

a b

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A

B C D

E

(C,a=1,b=1)

(D,a=1)(B,b=1)

(E,a=1)(C,a=1,b=0)

Example

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A

B C D

E

(C,a=1,b=1)

(D,a=1)(B,b=1)

(E,a=1)(C,a=1,b=0)

(B,b=0) (D,a=1)

Example

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A

C D

E

(C,a=1,b=1)

(D,a=1)(B,b=1)

(E,a=1)(C,a=1,b=0)

(B,b=0) (D,a=1)Much smaller search tree than with grid-based discretization !

B

Example

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Visible

Cleared

Contaminated

Example of Target-Finding Strategy

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More Complex Example

1

2

3

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Example with Recontaminations

654

321

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Example with Linear Number of

RecontaminationsRecontaminated area

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Example with Two Robots(Greedy algorithm)

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Example with Two Robots

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Example with Three Robots

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Robot with Cone of Vision

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Other Topics

Dimensioned targets and robots, three-dimensional environments

Non-guaranteed strategies Concurrent model construction and

target finding Planning the escape strategy of the

target