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Complexity Issues in Robot Motion Planning

Complexity Issues in Robot Motion Planning. Elif Tosun MTH 353 Final Paper. Overview. Introduction & Motivation Basic Definitions Basic Motion Planning Manipulation Planning Algorithmic Approaches Conclusions. Introduction.

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Complexity Issues in Robot Motion Planning

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  1. Complexity Issues in Robot Motion Planning Elif Tosun MTH 353 Final Paper

  2. Overview • Introduction & Motivation • Basic Definitions • Basic Motion Planning • Manipulation Planning • Algorithmic Approaches • Conclusions

  3. Introduction • Motion planning is aimed at providing robots with the capability of deciding automatically which motions to execute in order to achieve their tasks without colliding with other objects in their work space. • Considerations • Economic Cost - time, energy, etc. • Physical limitations - friction, etc.

  4. Motivation • Robot Applications • Manufacturing • Medical Surgery • Molecular Biology • Computer Graphics • Air & Spacecraft Navigation • ...

  5. Basic Definitions • Robot: Mechanical system consisting of one or more rigid bodies possibly connected by various joints and hinges • Configuration: Position of every point of a robot at a given instance • Degrees of Freedom(DOF): number of dimensions along which the robot can move itself.

  6. More definitions • Workspace: Environment in which the robot moves (2D or 3D) • Configuration space: Space of all configurations of a robot • Free space: configuration space minus the space occupied by obstacles

  7. Basic Motion Planning Objective: To plan a collision free path of a robot with an arbitrary DOF to a goal position in 2D or 3D avoiding a set of obstacles stationary in space.

  8. Complexity Results • 3D version : Robot is a set of linked polyhedra and obstacles are fixed polyhedra in 3D • Complexity: • PSPACE-hard when the robot has n links (due to Reif, proved in 1979) • P when the robot has a constant DOF (due to Schwartz & Sharir, proved in 1983)

  9. Complexity Results • 2D version : Robot is a set of linked polygons and obstacles are fixed polygonals objects in 2D • Complexity: • PSPACE-hard lower bound (due to Schwartz & Sharir, proved in 1984)

  10. Manipulation Planning • Objective: To have the robot move around objects in the workspace to reach a final arrangement. (Objects cannot move by themselves) • Games: • SOKOBAN • PushPush

  11. Sokoban • Objective of Robot: To push boxes into their storage locations without getting himself or boxes stuck. • Rules: Cannot pull, can push only one box at a time

  12. Sokoban • Complexity Result: Proved to be PSPACE-hard So all puzzles of this kind (different levels, etc.) are PSPACE-complete (due to Culberson, 1998)

  13. PushPush • Objective: To push blocks in order to get from an initial position to a final position • Rules: -One block at a time -Block slides the full extent of available space

  14. PushPush • Complexity Result • PushPush is NP-hard in 2D and 3D • Proof based on reduction from SAT. • Open question • Is it NP-Complete (is it in NP?) OR • is it PSPACE-Complete??

  15. Algorithmic Approaches • Complete Algorithms • Probabilistic Algorithms • Heuristic Algorithms

  16. Complete Algorithms • Guaranteed to find a free path between two give configurations when exists and report failure otherwise • Deal with connectivity of free space by capturing it on a graph. • Cell Decomposition - partition of free space • Roadmap Technique - network of curves • Not open for improvements

  17. Probabilistic Algorithms • Trade-off exactness against running time • Don’t guarantee a solution but if exists very likely to find it relatively quickly • Example: Probabilistic Roadmap Algorithm • Experimental results show that computation takes less than a second • Used in maintenance of aircraft

  18. Heuristic Algorithms • Many work well in practice but offer no performance guarantee • Deal with a grid on configuration space • Example 1 : Potential Field • Example 2 : Approximate Cell Decomposition • Space for Improvement

  19. Conclusions • Robot Motion Planning is DIFFICULT!!! • Many open problems: • motion planning with uncertainty • assembly planning • approximation algorithms • motion with flexible objects, and many more... • Interest moving from theoretical research to approximation algorithms and applications

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