introduction to robotics tutorial 10 technion, cs department, introduction to robotics 236927 winter...
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Destination 3 The destination is modeled as an attractive charge. DestinationTRANSCRIPT
Introduction to Robotics
Tutorial 10
Technion, cs department, Introduction to Robotics 236927
Winter 2012-2013
1
Potential Functions
2
1. Write the attraction and repulsion potential functions.
Destination
ObstacleCenter at (L,0)Radius = R
x
y
Destination
3
• The destination is modeled as an attractive charge.
Destination
x
y 22, yxyxU
ddUA
A
-10-5
05
10
-10-5
05
100
5
10
15
-10 -5 0 5 10-10
-8
-6
-4
-2
0
2
4
6
8
10
Potential Functions
4
Gradient Descent• Gradient descent is a well-known
approach to optimization problems. The idea is simple. Starting at the initial configuration, take a small step in the direction opposite the gradient. This gives a new configuration, and the process is repeated until the gradient is zero. More formally, we can define a gradient descent algorithm
Gradient Descent
Obstacle
7
• The Obstacle is modeled as a single repulsive charge.
22/,
/
yLxyxU
ddUR
R
ObstacleCenter at (L,0)Radius = R
x
y
-10-5
05
10
-10
-5
0
5
100
5
10
15
-10 -5 0 5 10-10
-8
-6
-4
-2
0
2
4
6
8
10
Obstacle and Destination
8
yxUyxUyxU RA ,,,
-10-5
05
10
-10-5
05
100
5
10
15
20
25
30
Obstacle and Destination
9
yxFyxFyxF RA ,,,
-10 -5 0 5 10-10
-8
-6
-4
-2
0
2
4
6
8
10
2 3 4 5 6 7 8-3
-2
-1
0
1
2
3
Potential Functions
10
2. For which α and β the robot will never hit the obstacle?
yyxLx
yLxyyxx
yx
yxFyxFyxF RA
ˆˆˆˆ
,,,
2/32222
Destination
ObstacleCenter at (L,0)Radius = R
x
y
Potential Functions
11
3. Will the robot always arrive at the destination?
4. From which starting positions the robot will not arrive the destination?
Local Minima Problem
Different Obstacle Modeling
13
• The Obstacle is modeled as a single repulsive charge.
22
0/
yLxd
elseRdRd
dU R
Potential Functions
14
5. For which α and β the robot will never hit the obstacle?
6. Will the robot always arrive at the destination?
7. From which starting positions the robot will not arrive the destination?
8. How does changing β effects the resulting path?
Different Obstacle Modeling
15
• The Obstacle is modeled as a single repulsive charge:
• Alternately:
Where d* is the distance to the closest point of the obstacle.
22
0/
yLxd
elseRdRd
dU R
elsedd
dU R
00/ **
*
Different Obstacle Modeling
16
elsedd
dU R
00/ **
*
Another Example
17
Destination
x
y