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

2. Overview zIntroduction & Motivation zBasic Definitions zBasic Motion Planning zManipulation Planning zAlgorithmic Approaches zConclusions

3. Introduction zMotion 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. zConsiderations yEconomic Cost - time, energy, etc. yPhysical limitations - friction, etc.

4. Motivation zRobot Applications zManufacturing zMedical Surgery zMolecular Biology zComputer Graphics zAir & Spacecraft Navigation z...

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

6. More definitions zWorkspace: Environment in which the robot moves (2D or 3D) zConfiguration space: Space of all configurations of a robot zFree 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 z3D version : Robot is a set of linked polyhedra and obstacles are fixed polyhedra in 3D zComplexity: yPSPACE-hard when the robot has n links (due to Reif, proved in 1979) yP when the robot has a constant DOF (due to Schwartz & Sharir, proved in 1983)

9. Complexity Results z2D version : Robot is a set of linked polygons and obstacles are fixed polygonals objects in 2D zComplexity: yPSPACE-hard lower bound (due to Schwartz & Sharir, proved in 1984)

10. Manipulation Planning zObjective: To have the robot move around objects in the workspace to reach a final arrangement. (Objects cannot move by themselves) zGames: ySOKOBAN yPushPush

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

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

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

14. PushPush zComplexity Result yPushPush is NP-hard in 2D and 3D yProof based on reduction from SAT. yOpen question xIs it NP-Complete (is it in NP?) OR xis it PSPACE-Complete??

15. Algorithmic Approaches zComplete Algorithms zProbabilistic Algorithms zHeuristic Algorithms

16. Complete Algorithms zGuaranteed to find a free path between two give configurations when exists and report failure otherwise zDeal with connectivity of free space by capturing it on a graph. yCell Decomposition - partition of free space yRoadmap Technique - network of curves zNot open for improvements

17. Probabilistic Algorithms zTrade-off exactness against running time zDon’t guarantee a solution but if exists very likely to find it relatively quickly zExample: Probabilistic Roadmap Algorithm zExperimental results show that computation takes less than a second zUsed in maintenance of aircraft

18. Heuristic Algorithms zMany work well in practice but offer no performance guarantee zDeal with a grid on configuration space yExample 1 : Potential Field yExample 2 : Approximate Cell Decomposition zSpace for Improvement

19. Conclusions zRobot Motion Planning is DIFFICULT!!! zMany open problems: ymotion planning with uncertainty yassembly planning yapproximation algorithms ymotion with flexible objects, and many more... zInterest moving from theoretical research to approximation algorithms and applications