CS221 Problem Set #1
1
CS 221
Problem Set #1: Search, Motion Planning, CSPs
Due by 9:30am on Tuesday, January 27. Please see the course information page on
the class website for late homework submission instructions.
SCPD students can also fax their solutions to (650) 7251449. We will not accept
solutions by email or courier.
Written part (100 points)
NOTE: These questions require thought, but do not require long answers. Please try to be as
concise as possible.
1.
[12 points] Configuration Spaces
d
robot
arm
floor
ceiling
base
gripper
ρ
α
Consider the robot arm pictured above with two degrees of freedom, operating in a two
dimensional workspace.
The robot arm has a revolute joint and a prismatic joint.
The
revolute joint has a range of 0
≤
α
≤
π
, where
α
is the angle of the arm relative to the floor.
The prismatic joint has a range of
ρ
min
≤
ρ
≤
ρ
max
, where
ρ
is the length of the arm from
the base to the gripper. The ceiling is a distance
d
from the floor, with
ρ
min
< d < ρ
max
.
The width of the arm and gripper may be considered negligible.
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CS221 Problem Set #1
2
θ
a
o
h
=o/h
θ
sin
cos
θ
=a/h
Draw the configuration space of the robot arm,
using
α
and
ρ
as the configuration space
coordinates
.
Please specify the coordinates of the important points of the obstacles in configuration
space (e.g.
leftmost point, rightmost point, etc.).
Also, while a freehand drawing is ac
ceptable, please make sure that the shape of the obstacle is clear from your drawing.
2.
[26 points] Optimization Search / Configuration Spaces
For this problem, we consider the problem of moving a robot in various configuration spaces
with obstacles.
We start with a simple configuration space, with only convex obstacles,
and then make it more complicated by allowing nonconvex obstacles as well.
convex obstacle
nonconvex obstacle
A discrete search space for this planning problem can be defined using the
visibility graph
method
. In this method, the search space is restricted to consist of only the following states
in configuration space: the initial position of the robot, the goal position of the robot, and
the
vertices
of the polygonal obstacles. Further, the search operators only allow the robot
to walk in a straight line between two of these points: a state
s
in the search space is
connected to any other state
s
′
which can be reached from
s
by walking along a straight
line either completely in free space or along the boundary of an obstacle (i.e.
s
is connected
to
s
′
if
s
′
is “visible” from
s
).
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 '09
 KOLLER,NG
 Optimization, search space, Constraint satisfaction, heuristic function, Admissible heuristic, Consistent heuristic

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