1. Introduction to Robotics Tutorial 10 Technion, cs department, Introduction to Robotics 236927 Winter 2012-2013 1

2. Potential Functions 2 1.Write the attraction and repulsion potential functions. Destination Obstacle Center at (L,0) Radius = R

3. Destination 3 The destination is modeled as an attractive charge. Destination

4. Potential Functions 4

5. 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

6. Gradient Descent

7. Obstacle 7 The Obstacle is modeled as a single repulsive charge. Obstacle Center at (L,0) Radius = R

8. Obstacle and Destination 8

9. 9

10. Potential Functions 10 2.For which α and β the robot will never hit the obstacle? Destination Obstacle Center at (L,0) Radius = R

11. Potential Functions 11 3.Will the robot always arrive at the destination? 4.From which starting positions the robot will not arrive the destination?

12. Local Minima Problem

13. Different Obstacle Modeling 13 The Obstacle is modeled as a single repulsive charge.

14. 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?

15. 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.

16. Different Obstacle Modeling 16

17. Another Example 17 Destination