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Motion Planning
(It’s all in the discretization)
R&N: Chap. 25 gives some background
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Motion planning
is the ability for an agent to
compute its own motions in order to achieve
certain goals. All autonomous robots and digital
actors should eventually have this ability
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Digital Actors
video 1
video 2
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Basic problem
Point robot in a 2-dimensional workspace with
obstacles of known shape and position
Find a collision-free path between a start and
a goal position of the robot
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Basic problem
Each robot position (x,y) can be seen as a state
→
Continuous
state space
Then each state has an infinity of successors
We need to
discretize
the state space
(x,y)
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Two Possible Discretizations
Grid-based
Criticality-based

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Two Possible Discretizations
Grid-based
Criticality-based
But this problem is very simple
How do these discretizations scale up?
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Intruder Finding Problem
A moving intruder is hiding in a 2-D workspace
The robot must “sweep” the workspace to find
the intruder
Both the robot and the intruder are points
robot’s
visibility
region
hiding
region
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cleared region
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3
4
5
6
robot
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Does a solution always exist?
Easy to test:
“Hole” in the workspace
Hard to test:
No “hole” in the workspace
No !
Information State
Example of an information state = (x,y,a=1,b=1,c=0)
An
initial state
is of the form (x,y,1, 1, ..., 1)
A
goal state
is any state of the form (x,y,0,0, ..., 0)
(x,y)
a = 0 or 1
c = 0 or 1
b = 0 or 1
0
Æ
cleared region
1
Æ
hidding region
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Critical Line
a=0
b=1
a=0
b=1
Information state is unchanged
a=0
b=0
Critical line
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A
B
C
D
E
Criticality-Based Discretization
Each of the regions A, B, C, D, and E
consists of “equivalent” positions of the robot,
so it’s sufficient to consider a single position
per region

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Criticality-Based Discretization
A
B
C
D
E
(C, 1, 1)
(D, 1)
(B, 1)
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Criticality-Based Discretization
A
B
C
D
E
(C, 1, 1)
(D, 1)
(B, 1)
(E, 1)
(C, 1, 0)
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Criticality-Based Discretization
A
B
C
D
E
(C, 1, 1)
(D, 1)
(B, 1)
(E, 1)
(C, 1, 0)
(B, 0)
(D, 1)
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Criticality-Based Discretization
A
C
D
E
(C, 1, 1)
(D, 1)
(B, 1)
(E, 1)
(C, 1, 0)
(B, 0)
(D, 1)
Much smaller search tree than
with grid-based discretization !
B
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Grid-Based Discretization
Ignores critical lines
Æ
Visits many “equivalent” states
Many information states per grid point
Potentially very inefficient
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Example of Solution

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