1. Configuration Spaces for Translating Robots Minkowsi Sum/Difference David Johnson

2. C-Obstacles Convert – robot and obstacles – point and configuration space obstacles Workspace robot and obstacle C-space robot and obstacle

3. Translating Robots Most C-obstacles have mysterious form Special case for translating robots Look at the 1D case -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 robot obstacle

4. Translating Robots What translations of the robot result in a collision? -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 robot obstacle

5. Minkowski Difference The red C-obs is the Minkowski difference of the robot and the obstacle -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 robot obstacle

6. Minkowski Sum First, let us define the Minkowski Sum

7. Minkowski Sum A B

11. Minkowski Sum Example Applet The Minkowski sum is like a convolution A related operation produces the C-obs – Minkowski difference

12. Back to the 1D Example What translations of the robot result in a collision? -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 robot obstacle

13. Tracing Out Collision Possibilities

14. Minkowski Difference -B

15. From sets to polygons Set definitions are not very practical/implementable For polygons, only need to consider vertices – Computationally tractable

16. Properties of Minkowski Difference For obstacle O and robot R – if O - R contains the origin Collision!

17. Another property The closest point on the Minkowski difference to the origin is the distance between polygons Distance between polygons

18. Example Applet

19. Discussion Given a polygonal, translating robot Polygonal obstacles Compute exact configuration space obstacle Next class – how will we use this to make paths?