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
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
robotobstacle
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
robotobstacle
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
robotobstacle
Minkowski Sum
• First, let us define the Minkowski Sum
Minkowski Sum
},|{ BbAabaBA
A B
Minkowski Sum
}|{ AaBaBA },|{ BbAabaBA
Minkowski Sum
Minkowski Sum
},|{ BbAabaBA
Minkowski Sum Example
},|{ BbAabaBA
• 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?
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
robotobstacle
},|{ RbOabaRO
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
0
0
RyOy
RxOx
• For obstacle O and robot R– if O - R contains the origin
Collision!
0,0 RptOpt
Ry
Rx
Oy
Ox
RptOpt
Another property
BbAabaBAd ,,min),(
• The closest point on the Minkowski difference to the origin is the distance between polygons
• Distance between polygons
BAzzBAd ,min),(
Discussion
• Given a polygonal, translating robot• Polygonal obstacles
• Compute exact configuration space obstacle
• Next class – how will we use this to make paths?
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