Configuration Spaces for Translating Robots Minkowsi Sum/Difference David Johnson

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

Translating Robots Most C-obstacles have mysterious form Special case for translating robots Look at the 1D case robot obstacle

Translating Robots What translations of the robot result in a collision? robot obstacle

Minkowski Difference The red C-obs is the Minkowski difference of the robot and the obstacle robot obstacle

Minkowski Sum First, let us define the Minkowski Sum

Minkowski Sum A B

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

Back to the 1D Example What translations of the robot result in a collision? robot obstacle

Tracing Out Collision Possibilities

Minkowski Difference -B

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

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

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

Example Applet

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