CS221 Problem Set #1
1
CS 221
Problem Set #1: Search, Motion Planning, CSPs
Due by 9:30am on Tuesday, October 13. 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 (70 points)
NOTE: These questions require thought, but do not require long answers. Please try to be as
concise as possible.
1.
[10 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 coordinates of the
configuration space
(i.e., the horizontal and verical axes in your figure should be labeled
α
and
ρ
).
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.
[14 points] Optimization Search / Configuration Spaces
We consider the problem of moving a robot in various two dimensional (2D) configura
tion 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, we place landmarks in configuration space at 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.,
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 Fall '09
 KOLLER,NG
 Optimization, search space, Constraint satisfaction, heuristic function, Admissible heuristic, Consistent heuristic

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