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Configuration Spaces for Translating Robots Minkowsi Sum/Difference

Configuration Spaces for Translating Robots Minkowsi Sum/Difference. David Johnson. C-Obstacles. Convert robot and obstacles point and configuration space obstacles. C-space robot and obstacle. Workspace robot and obstacle. Translating Robots. Most C-obstacles have mysterious form

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Configuration Spaces for Translating Robots Minkowsi Sum/Difference

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  1. Configuration Spaces for Translating RobotsMinkowsi Sum/Difference David Johnson

  2. C-Obstacles • Convert • robot and obstacles • point and configuration space obstacles C-space robot and obstacle Workspace 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 B A

  8. Minkowski Sum

  9. Minkowski Sum

  10. Minkowski Sum

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

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

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

  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?

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