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